Signal processor with local signal behavior

ABSTRACT

A new signal processing method and a signal processing engine which can achieve extremely fast responsiveness to instantaneous changes in the behavior of the signal, and maintain the accuracy of standard harmonic methods. The signal processing engine unifies Nyquist&#39;s theorem and Taylor&#39;s theorem by means of polynomial approximations using linear operators, e.g. differential and integral operators. The signal processing engine samples the signal at a rate which is n times the band limit of the signal, where n is greater than 2, i.e. greater than the Nyquist rate, produces a digital representation of the sampled signal, and calculates the outputs of linear operators applied to polynomial approximations of the sampled signal. A switch mode power amplifier which incorporates the signal processing method and engine of the overcomes shortcomings of existing switching amplifiers, e.g. class &#34;D&#34; amplifiers. These shortcomings include: poor handling of highly reactive complex loads (e.g., speakers), usually requiring a duty cycle or feed-back adjustment with the change of the load; poor performance in the upper part of the bandwidth, including numerous switching artifacts; and high distortion, especially in the upper part of the spectrum. These shortcomings are all overcome using the local signal behavior signal processing method and engine of the invention.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority from my copending provisional U.S.patent applications 60/061,109, filed Oct. 3, 1997, and 60/087,006,filed May 28, 1998, the disclosures of both of which are incorporatedherein by reference.

FIELD OF THE INVENTION

This invention relates to signal processing, and particularly to amethod and apparatus for highly accurate real-time signal processingwith essentially no delay. More particularly, the invention relates to asignal processing method and apparatus having predictive features whichallow extremely fast responsiveness to instantaneous changes in thebehavior of a signal.

BACKGROUND OF THE INVENTION

Numerical harmonic analysis was developed before the advent of computingmachines for the purpose of analyzing motions which can be adequatelyrepresented as compositions of harmonic oscillations. It was importantthat the analysis methods were as efficient as possible because thecomputations were done by hand. In this respect, Fourier analysis is thefastest method which uses the minimal amount of data to analyze suchsignals. Also, the first digital computers and analog to digital (A/D)converters had limited speed and resolution, so the same methods used inmanual computation were developed for automated computation. In recentyears the processing speeds of computers, and speed and accuracy of A/Dconverters have dramatically improved. However, the same harmonicanalysis methods which predated the advances in computing devices andA/D converters are still in dominant use. Harmonic methods were notintended for use in conditions requiring extremely small processingdelay, and are deficient when used under such conditions for intrinsicreasons.

Present day digital signal processing relies primarily oncomputationally intensive Fourier analysis based methods which use arelatively large number of samples taken at the Nyquist rate. Such longsequences of samples encode the "global" behavior of the input signal,i.e., as used herein, the behavior of the signal over a relatively longperiod of time lasting many Nyquist rate sampling intervals. However,when the signal is represented by such values, sampled at the Nyquistrate, any direct access to the information about the "local" behavior ofthe input signal, i.e., as used herein, the behavior of the signal overshort intervals of time, in a neighborhood between two consecutiveNyquist rate samples, is lost.

Harmonic analysis represents the signal using trigonometric functions.Local variations of a signal in time are very poorly represented byperiodic functions which are highly uniform in time and are suitableonly for global representations of the signal over longer periods oftime, i.e., many tens, or hundreds or even thousands of Nyquist rateintervals.

Not having accurate information about the local behavior of a signalmakes it difficult to act in real time because, at any given instant oftime, action can only be based upon the instantaneous value of thesignal and values from the past. Due to the uncertainty principle,harmonic analysis requires that the signal be "seen" over a relativelylong period of time, i.e. a large number of consecutive Nyquist ratesamples are needed, often windowed toward the ends of the interval.Thus, harmonic analysis can only be used in instances in which delayand/or phase shifts inherent in such methods may be tolerated.

Some of the most sophisticated real time applications currently usedrely on wavelet methods with cardinal splines, however, these methodsstill use signal processing operators which are similar to those used inharmonic analysis.

Moreover, while Nyquist's theorem enables a complete representation of aband-limited signal using only the values of the signal sampled at theNyquist rate, if the signal is given by values sampled at the Nyquistrate only up to a time t₀, then no past values between the samplingtimes are fully determined. If the value of the signal at a time t isapproximated using the Nyquist rate values up to time t₀ >t, then theerror of the interpolation (oversampling) depends on the energy of thesignal contained in its part after t₀, and no a-priori bound can begiven only in terms of t₀ -t, i.e. the number of samples between times tand t₀. This problem is usually solved by replacing the original signalby an (overlapping) sequence of signals obtained by windowing theoriginal signal over a sufficiently long interval of time, thusrestricting the number of samples on which the interpolated values candepend.

On the other hand, Taylor's theorem provides all past and future valuesof the signal from the values of all derivatives of the signal at asingle instant in time t₀. However, Taylor's theorem is difficult to usein practice because the higher order derivatives, being extremely noisesensitive, cannot be evaluated precisely. Also, the truncated Taylorformula quickly accumulates error moving away from the point ofexpansion, t₀. Taylor's theorem implies that the signal is determined byall of its values in the past (i.e., not just the Nyquist rate values,but values at every instant). This determinism makes such a model, evenif it were feasible, inadequate for practice.

There is a need for a signal processing method and a signal processorwhich can characterize the local behavior of a band limited signal interms of some suitable parameters, and which can relate such signals'local behavior parameters to the spectrum of the signal, and thus alsoto the global behavior of the signal. Such a method should provide acomputationally effective way of achieving extremely fast responsivenessto instantaneous changes in the behavior of the signal while maintainingthe spectral accuracy achieved with standard harmonic methods.

SUMMARY OF THE INVENTION

The present invention provides a new signal processing method and asignal processor, referred to herein as the "engine," which can achieveextremely fast responsiveness to instantaneous changes in the behaviorof the signal, and maintain the spectral accuracy of standard harmonicmethods. The signal processing engine of the invention is usefulprimarily, but not exclusively, in real time applications.

A main feature of the signal processing method of the invention is inthe manner by which it reconciles the limitations imposed by theUncertainty Principle on low delay signal processing. Observing a signalover a short period of time implies insufficient accuracy of therepresentation of the signal's spectral content. However, in the methodof the invention, this is overcome by a powerful aliasing principle (aFinite Local ε-base Theorem, set forth herein below) which allows localsignal processing based on "spectral consistency" rather than absolutespectral accuracy, by locally defined operators, by a non-uniformrepresentation of the spectrum of the signal which is more detailed forhigher frequencies, and which require shorter windows for adequateresolution. These features allow local operations which "anticipate" thefuture values of the signal by unifying two paradigms, Nyquist's theoremand Taylor's theorem, using polynomial approximations of the inputsignal, and of the outputs of linear operators applied to polynomialapproximations of the input signal, most notably various differentialoperators. The predicted values are fully adequate for computingparameters describing local signal behavior at the very end of the datastream of sampled values of the signal.

In a first aspect, the invention provides a signal processor comprisingdata description means for characterizing the local behavior of a bandlimited signal, the data description means comprising data acquisitionmeans for sampling the signal at a rate which is n times the band limitof the signal, where n is greater than 2, and local signal behaviordescriptor means for calculating the output of a linear operator appliedto a polynomial approximation of the sampled signal. This representationcan be either in a digital or in an analog format.

As used herein, the term "oversampling" refers to sampling a signal at arate which is n times the band limit of the signal, where n is greaterthan 2, i.e., greater than the Nyquist rate. Since it is common toselect sampling rates which are multiples of powers of 2, multiples ofpowers of 10, convenient components of available clock rates, and thelike, it is understood that, as used herein, samples taken at theNyquist rate may be at a rate which is at least, but not necessarilyexactly twice the band limit of the signal.

The method of the invention is a form of a multirate signal processing.From the sequence of oversampled data, a Nyquist rate subsequence isselected, and this subsequence and the total oversampled data are usedwith very different roles both methodologically and computationally. Ina multi-layered local signal behavior (LSB) method, sub-Nyquist ratesequences are also selected and used in a similar manner.

In a second aspect, the invention provides a signal processing methodfor characterizing the local behavior of a band limited signal, thesignal processing method comprising the steps of sampling the signal ata rate which is n times the band limit of the signal, where n is greaterthan 2, and calculating the output of a linear operator applied to apolynomial approximation of the sampled signal.

In a third aspect, the invention provides a signal processing method forcharacterizing the local behavior of a band limited signal, the signalprocessing method comprising the steps of sampling the signal at a ratewhich is m times the Nyquist rate for the band limit of the signal,where m is greater than 1, and calculating a local signal descriptionparameter for a sampled value of the signal at a time t₀, the localsignal description parameter comprising the output of a linear operatorapplied to a polynomial approximation of the signal at the time t₀, thepolynomial approximation comprising not more than about 12 to 24 Nyquistrate samples of the signal and substantially all of the samples of thesignal from not more than about 1 to 5 Nyquist rate intervals.

In preferred embodiments, the linear operator comprises either a locallysupported operator or an operator defined by recursion from locallysupported operators, which may be a differential operator, an integraloperator, an interpolation operator, an extrapolation operator or acombination of these operators.

In preferred embodiments, the invention provides a new class ofdifferential operators, identified herein as chromatic derivativesbecause they encode the spectral features of the signal.

In preferred embodiments of the invention, the signal processing methodand engine employ a polynomial approximation of an input signal whichcomprises a Lagrange polynomial approximation of the signal.

In preferred embodiments of the invention, the signal processing methodand engine employ a polynomial approximation of an input signal whichcomprises a piecewise polynomial approximation of the signal.

In preferred embodiments of the invention, the signal processing methodand engine may employ a transversal filter to implement a polynomialapproximation of an input signal, and may employ a transversal filter toimplement a linear operator applied to a polynomial approximation of theinput signal.

In preferred embodiments of the invention, the data acquisition meansemploys a method which naturally rejects the out of band noise,including the quantization noise, and an adaptive technique forestablishing the dynamic range of the A/D conversion. The dataacquisition means has settings for full scale and resolution. Takingadvantage of the predictive capabilities of the invention, a predictedvalue of the sampled signal is determined, the difference between thevalue of the sampled signal and the predicted value of the sampledsignal are calculated, and the full scale and the resolution areadjusted in response to the difference between the value of the sampledsignal and the predicted value of the sampled signal. In preferredembodiments, the data acquisition means also has a soft-start capabilityfor dampening initialization transients.

In a fourth aspect, the invention provides switch mode amplifiercomprising input means for receiving an input voltage, a low pass filterfor providing an output voltage and output current to the load, aswitching regulator for regulating a voltage input to the low passfilter, pulse width modulation control means for controlling theswitching regulator, correction means for comparing the output voltagewith the input voltage and supplying a correction current to the loadbased on the results of the comparison, an inner feedback loop forsensing a switching regulator output voltage at an output of theswitching regulator and providing an inner feedback loop input to thepulse width modulation control means, the inner feedback loop inputresponsive to the switching regulator output voltage, the inner feedbackloop comprising a first signal processor according to the first aspectof the invention, the first signal processor adapted to input theswitching regulator output voltage and providing the audible componentof the switching regular output voltage, an outer feedback loop forsensing the output current and the correction current and providing anouter feedback loop input to the pulse width modulation control means,the outer feedback loop input responsive to the output current and thecorrection current, the outer feedback loop comprising a second signalprocessor according to first aspect of the invention, the second signalprocessor adapted to input the sensed value of the correction currentand to output a first derivative of the correction current, and a thirdsignal processor according to the first aspect of the invention, thethird signal processor adapted to input the sensed value of the outputcurrent, and to output a first derivative of the output current, and afourth signal processor according to the first aspect of the invention,the fourth signal processor adapted to receive the input voltage and tooutput a second derivative of the input voltage to the pulse widthmodulation control means.

The present invention provides not merely a signal processing algorithm,but rather an entirely new approach to signal processing and an entirelynew signal processing technology. The signal processing method andengine of the invention can be used to advantage in communications,analog to digital and digital to analog conversion, signal encoding,control, power electronics, signal compression, digital image encodingand processing, and empirical data processing and forecasting ineconomics and the social sciences.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph which illustrates the nature and a limitation ofNyquist's theorem.

FIG. 2 is a graph which illustrates the nature and a limitation ofTaylor's theorem.

FIG. 3 compares a graph of the sixth order derivative of (sin πx)/(πx)with a graph of a corresponding polynomial approximation of the sixthorder derivative of (sin πx)/(πx) with both graphs calculated andplotted by the program Mathematica.

FIGS. 4 and 5 are graphs which compare a polynomial approximation of(sin πx)/(πx) with the function (sin πx)/(πx) on two different scales.

FIG. 6 is a graph which illustrates the effects of windowing.

FIGS. 7A-7F are graphs which illustrate the error of approximation of(sin πx)/(πx) by the Lagrangian polynomial L₀ ⁴⁷ (t).

FIG. 8 is a diagram of an embodiment of a first general transversalfilter which may be used to compute the output of a linear operatorapplied to a polynomial approximation of a signal.

FIG. 9 is a diagram of an embodiment of a second general transversalfilter which may be used to compute the output of a linear operatorapplied to a polynomial approximation of a signal.

FIG. 10 is a flow chart of an embodiment of a routine which implementsthe first and second general transversal filters illustrated in FIGS. 8and 10.

FIG. 11 is a diagram of an embodiment of a third general transversalfilter which may be used to compute the outputs of a plurality of linearoperators applied to polynomial approximations of a signal.

FIG. 12 is a diagram of an embodiment of a fourth general transversalfilter which may be used to compute the outputs of a plurality of linearoperators applied to polynomial approximations of a signal.

FIG. 13 is a diagram of an embodiment of a fifth general transversalfilter which may be used to compute the output of a quadraticoptimization procedure involving a polynomial approximation of a signal.

FIG. 14 is a diagram of an embodiment of a transversal filter which maybe used to compute the m^(th) derivative of a polynomial approximationof a signal.

FIGS. 15A-15D are graphs of |ω/π|^(2j) for j=1, 2, 3, 4, 5, 6, 15 and16.

FIGS. 16A-16D are graphs of |T_(n) (ω/π)|² for n=0, 1, 2, 3, 4, 5 and16.

FIG. 17 is a diagram of an embodiment of a recursive infinite response(IIR) system performing integration of the input signal from a fixedpoint in time up to the time t_(j).

FIG. 18 is a graph of the characteristic of an embodiment of a Σ-Δfilter.

FIG. 19 is a diagram of an embodiment of a Σ-Δ filter which has thecharacteristic illustrated in FIG. 18.

FIG. 20 is a diagram of an embodiment of a Σ-Δ filter of FIG. 19 havingtwo accumulators.

FIG. 21 is a graph which illustrates the characteristics of a monad.

FIG. 22 is a graph which illustrates a link.

FIG. 23 is a graph which illustrates a complex having 5 links.

FIG. 24 is a graph which illustrates an intermediate complex.

FIG. 25 is a graph which illustrates a fully predictive complex.

FIG. 26 is a block diagram illustrating the top level components of anembodiment of a signal processing engine according to a first aspect ofthe invention.

FIG. 27 is a block diagram illustrating the components of an embodimentof a data acquisition unit of a signal processing engine according to afirst aspect of the invention.

FIG. 28 is a block diagram illustrating the components of an embodimentof a prediction filter of a signal processing engine according to afirst aspect of the invention.

FIGS. 29A and 29B are flow charts which illustrate aspects of theoperation of an embodiment of a control logic module of a signalprocessing engine according to a first aspect of the invention.

FIG. 30 is a block diagram illustrating the components of a secondembodiment of a data acquisition unit of a signal processing engineaccording to a first aspect of the invention.

FIG. 31 is a graph illustrating the transfer function of an embodimentof a soft-start window of a signal processing engine according to afirst aspect of the invention.

FIG. 32 is a flow chart which illustrates the transient recoveryoperation of the control logic in the second embodiment of a dataacquisition unit illustrated in FIG. 30.

FIG. 33 is a block diagram illustrating components of an embodiment of aLocal Signal Behavior Descriptor according to the invention.

FIG. 34 is a diagram which illustrates how filtering can be implementedin a mix of local signal behavior plus harmonic technologies.

FIG. 35 is a diagram illustrating an embodiment of a switch mode poweramplifier employing the signal processing engine of the invention.

FIG. 36 is a circuit diagram of certain components of the switch modeamplifier illustrated in FIG. 35.

FIG. 37 is a diagram of an embodiment of a transversal filter suitablefor use in LSB Engine 2 and LSB Engine 3 in the switch mode amplifierillustrated in FIG. 35.

FIG. 38 is a diagram of a portion of the transversal filter illustratedin FIG. 37 with an analog leaky integrator coupled to the input of thefilter.

FIG. 39 is a diagram of an embodiment of a transversal filter suitablefor use in LSB Engine 4 in the switch mode amplifier illustrated in FIG.35.

FIGS. 40, 41 and 42 are diagrams which illustrate differences betweenthe approach used by harmonic analysis and the approach used by thelocal signal behavior processing method and engine of the invention.

DETAILED DESCRIPTION OF THE INVENTION

The signal processing method and engine of the invention represent anentirely new signal processing technology. In order to present adetailed description of preferred embodiments of the invention, it isappropriate to define certain principles of the mathematical basis ofthe invention, and certain operations employed by the invention. Thefollowing description includes the mathematical principles of theinvention as they pertain to preferred embodiments of the signalprocessing method and engine of the invention, and also includes adescription of an embodiment of a switch mode amplifier employing thesignal processing method and engine in accordance with the invention.

The detailed description of the invention is presented according to thefollowing outline:

I. Nyquist's Theorem

II. Taylor's Theorem

III. Polynomial Approximation and Differential and Integral Operators

A. Differential Operators

B. Integral Operators

C. Locally Supported Operators

D. Polynomial Approximations

E. Lagrange Approximation Polynomials

F. Polynomial Approximation Theorem

IV. Signal Processing Based on Lagrangian Approximation Polynomials

A. Approximation Polynomials Represented by Transversal Filters

B. Differentiation

C. Integration

D. Σ-Δ Procedures

E. Solving Linear Differential Equations Using Lagrangian PolynomialApproximations

V. Basic Signal Processing Local Universes

A. Finite Local ε-Base Theorem

B. k-Monad

C. p-Simplex

D. m-Complex, Sequences of Local Universes

VI. Local Signal Behavior Parameters

VII. Interpolation Polynomials in General

VIII. Derivation of Local Signal Behavior Processing Parameters

IX. Signal Processing Engine

X. Switch Mode Amplifier

XI. Summary: Harmonic Analysis and Local Signal Behavior Processing

If not stated explicitly otherwise, a unit interval of time is theinterval between two Nyquist rate sampling points.

I. Nyquist's Theorem

According to Nyquist's theorem, every band limited signal can bereconstructed from its samples taken at a rate equal to twice the bandlimit of the signal. However, as illustrated in FIG. 1, many samples areneeded to interpolate between the sampling points, e.g. at the point τ.FIG. 1 is a graph which illustrates the nature and a limitation ofNyquist's theorem. FIG. 1 is a graph of a band-limited signal f which isbeing sampled at the Nyquist rate. As indicated on the graph, in orderto determine the value of f at time τ, which falls between samplingtimes t₀ and t₁, it is necessary to obtain samples essentially "adinfinitum" in both directions.

The reconstruction formula is: ##EQU1##

In practice, many continuous signals can be entirely reconstructed fromtheir discrete samples, taken sufficiently frequently. However, signalprocessing based on Nyquist's theorem poses several problems because itrequires Nyquist rate samples from a very long interval of time, andintensive computations based on harmonic analysis. For example, in thereconstruction formula, the impact of the n^(th) term on the values ofthe signal around t=0 can be on the order of magnitude 1/(nπ). Thus, theimpact of the terms decreases only linearly in n. Since the series##EQU2## diverges, the Nyquist interpolation formula converges only byvirtue of the fact that f has finite energy, i.e., that ##EQU3##

This means that, in general, the values of the signal at very distantsampling points might be needed in order to reconstruct the signal evenwith limited accuracy and in a small interval around t₀. On the otherhand, if the reconstruction formula is truncated up to a fixed large n,then the approximation remains accurate over a significant centralinterval of time between the points -n and n. Thus, the local behaviorof the signal, i.e., its behavior over short intervals of time betweentwo consecutive Nyquist sampling times is poorly encoded by Nyquist ratesamples, while the global behavior is encoded accurately by Nyquist ratesamples.

The value of a derivative of a signal, f, at a sampling point, t₀, is alocal operation because the value of the derivative is entirelydetermined by values of f in an arbitrarily small neighborhood aroundt₀. However, since the local behavior of the signal is poorly encoded byNyquist rate samples, local operations including derivatives aredifficult to perform precisely on signals given at the Nyquist rateusing the standard methods which are based essentially on the Nyquistreconstruction formula. To obtain the values of such local operations bythe standard methods requires the use of samples, some of which are veryfar away from t₀. While possible in principle, clearly such calculationseasily accumulate error.

In general, operations like differentiation which depend on the valuesof the signal only in a small interval of time (referred to herein aslocally supported operators), cannot be accurately computed usingmethods which rely on a trigonometric representation of the signal. Suchrepresentations, as noted above, are responsible for the extremeinefficiency of harmonic analysis techniques when dealing with the localsignal behavior.

On the other hand, operations based on the global behavior of thesignal, e.g. taking the Fourier transform, or filtering in the frequencydomain, can be performed very accurately using standard, global methodsbased on Fourier analysis and representation of the signal usingtrigonometric functions.

II. Taylor's Theorem

Band limited signals have several important characteristics. They areinfinitely differentiable, and the n^(th) derivative of f(t) is denotedby f.sup.(n) (t). (As used herein, the first derivative of a function isdenoted by a "prime," e.g. f'(t), and the n^(th) derivative, where n>1,is denoted by the superscript, n, in parentheses, e.g. f.sup.(n) (t),f.sup.(2) (t), etc.) Also, it can be proven that the Taylor series of aband-limited signal converges everywhere to the value of the signal:##EQU4##

Since the value of a derivative of a signal at a point t₀ is defined asa limit, this value is uniquely determined by an arbitrarily smallneighborhood of the point t₀. This is illustrated in FIG. 2 whichillustrates the nature and a limitation of Taylor's theorem. FIG. 2 is agraph of a band-limited signal f. As indicated on the graph, in order todetermine the value of the derivative of f at time t₀, it is necessaryto obtain the values of the signal f(t) "ad infinitum" close to thepoint t₀.

Taylor's theorem can be important in practice because every band limitedsignal, f, can be entirely reconstructed from the values of all of itsderivatives at a single instant in time, t₀. Thus, since the localbehavior of f, i.e. the behavior of f in an arbitrarily small intervalaround to, determines the values of all derivatives of f at t₀, thebehavior of the entire signal f at any instant in time is completelydetermined by its behavior in an arbitrarily small interval around t₀.This may be contrasted with Nyquist's theorem, which, for an idealreconstruction, requires samples of the signal at infinitely manysampling instants, spread throughout the entire time.

However, due to noise and round-off error, it is difficult to determinehigh order derivatives with high accuracy. Also, while the localbehavior of a signal is represented much better with Taylor's theoremthan by Nyquist's theorem, global behavior is very poorly represented byTaylor's theorem since the error of the truncated Taylor's theorem up tosome n increases very fast as t-t₀. increases.

This error is given by the following formula:

Let ##EQU5##

Then ##EQU6##

Here E is the energy of the signal, i.e. ##EQU7##

Clearly, if (n+1)>(eπ|t-t₀ |), then (eπ|t-t₀ |)/(n+1)<1, and the errordrops rapidly as n further increases. Thus, n=9 provides an accuraterepresentation of f, within one unit interval on the left and one unitinterval on the right. To have the same accuracy within 2 Nyquist rateintervals requires Taylor's formula of degree 17. However, due to theinconvenient filtering features of differentiation, Taylor's formula isnot used in this invention in its original form. The derivatives arereplaced with certain linear differential operators which are muchbetter suited for expanding band limited signals.

Comparing Nyquist's theorem with Taylor's theorem, it is apparent thatthe global (Nyquist's) and local (Taylor's) paradigms suffer fromcomplementary problems. In the global paradigm, one must go too far awayfrom a point of interest, t₀, in order to accumulate enough informationfrom sampled values. In the local paradigm, one must get too close t₀ toin order to evaluate the derivatives at t₀.

I have determined that the two paradigms can be combined by means ofcertain linear operators, particularly certain differential operators,applied to polynomial approximations of signals, and by using certainquadratic minimizations.

III. Polynomial Approximations and Differential and Integral Operators

The signal processing method and engine of the invention usesignificantly modified versions of both the truncated Nyquist and Taylorformulas applied to band limited signals to allow real time processing.

A. Linear Differential Operators

For an input signal f(t), a differential operator D produces an outputsignal D(f), which is given by

    D(f)=a.sub.0 f+a.sub.1 f'+ . . . +a.sub.n f.sup.(n)

where a₀, a₁, . . . , a_(n) are real numbers.

B. Linear Integral Operators

For an input signal f(t), an integral operator I produces an outputsignal I(f), satisfying

    a.sub.0 (I(f))+a.sub.1 (I(f))'+ . . . +a.sub.n (I(f)).sup.(n) =f.

The differential and integral operators are inverses to each other, i.e.integral operators provide solutions to differential equationscorresponding to inverses of differential operators.

C. Locally Supported Operators

As defined herein, an operator F acting on band limited signals is alocally supported operator if the value [F(f)] at a point t₀, i.e.[F(f)](t₀), depends only on the values of f in the interval containingt₀, one unit length to the left and one unit length to the right, i.e.(t-1, t+1).

This means that the values of outputs of such operators at any point t₀can be determined from the values of the signal only in a small intervalaround the point t₀. Differentiation of any order, and integration overany interval contained within the interval (t₋₁, t₁) are examples oflocally supported operators. Integration procedures over longerintervals are defined using recursion, i.e. as infinite impulse response(IIR) operators.

D. Polynomial Approximations

Noise prevents differentiating a signal directly. For example, for a 60Hz signal from an AC power supply with an amplitude of 110 voltscontaminated by a 60 kHz switching noise of amplitude of only 1.1 mV,the 60 kHz noise component of the second derivative is 10 times higherthan the amplitude of the main 60 Hz component.

In order to be able to use differential operators for processingsignals, instead of relying on filtering out the noise, the inventionuses methods which are naturally frequency selective and noise robust.These methods use a special kind of polynomial approximation, derivedfrom Lagrangian interpolation polynomials, to collect the globalinformation contained in the Nyquist rate samples taken over an intervalof time T, where T is from several to several tens of Nyquist ratesampling intervals. These methods transform such global information intolocal information associated with the central subinterval C of T oflength one or two Nyquist rate unit intervals.

One of the main features of such polynomials is the notion of scalewhich they inherit from the Lagrangian interpolation polynomials whichare used to derive the polynomial approximations used. The scaleprovides filtering features of such polynomials which gives thenecessary noise robustness. A second important feature of suchpolynomials is that they naturally provide (essentially exponential)windowing of the signal without any additional multiplication of thesignal by a windowing function.

Polynomial approximations are extremely smooth. For example, round-offerror prevents accurate evaluation of the sixth order derivative of (sinπx)/(πx), even in double precision. FIG. 3 compares a graph of the sixthorder derivative of (sin πx)/(πx) with a graph of a correspondingpolynomial approximation of the sixth order derivative of (sin πx)/(πx).Both graphs were produced using the Mathematica program. It can be seenthat in the neighborhood of the origin, the calculations of the sixthorder derivative of (sin πx)/(πx) produce values which oscillateviolently. The oscillatory behavior of the graph of the sixth orderderivative of (sin πx)/(πx) is due entirely to the round-off error.Differentiation of the sixth order of (sin πx)/(πx) produces an error of2×10¹⁵ around 0 with double precision accuracy, while a polynomialapproximation is perfectly stable. Despite the fact that, from theanalytical point of view, (sin πx)/(πx) is infinitely differentiable at0, from the point of view of numerical analysis, this function has a"singularity" at 0. Since for x=0 the denominator vanishes, and sincedifferentiation increases the power of the denominator, the fact thatthe n^(th) derivative of (sin πx)/(πx) converges becomes grosslyaffected even by small round-off errors of double precision.

The sixth order derivative of the polynomial approximation of (sinπx)/(πx) used in the method and signal processing engine of theinvention is accurate to about one percent of the full scale of thesixth order derivative of (sin πx)/(πx), and is perfectly stable andnearly insensitive to the round-off noise due to the large (Nyquistrate) spacing of interpolation points. This is an important feature ofthe Nyquist rate-spaced Lagrangian interpolation polynomials used in themethod and engine of the invention.

When a signal is oversampled, the method and signal processing engine ofthe invention use Lagrangian approximation polynomials to capture theshape of the signal in a small interval around a sampling point. This isdone by a special procedure using a quadratic optimization. Applyingdifferential and integral operators can be made with exceedingly highaccuracy, even at the end points of the data stream which explains whythe method of the invention is the best method for real-time signalprocessing.

E. Lagrange Approximation Polynomials

Consider the standard Lagrangian approximation polynomial constructedover a set of sampling points. With n sampling times, t₁, t₂, . . . ,t_(n), and the corresponding sampled values of the input signal f(t₁),f(t₂), . . . , f(t_(n)), then, with the variable t representing time,##EQU8## The polynomial, in the variable t ##EQU9## is the Lagrangeinterpolation polynomial, and L_(i) (t) are the Lagrange coefficients.The superscript, n, is omitted whenever possible. The interpolatingpolynomial obtained from n sampled values f(t_(i)) of the input signalf(t) at the sampling times t_(i) for 1≦i≦n, is of degree n-1.

The values of L_(i) (t) depend only on the spacing of the sampling timesand the spacing of the time t of interpolation (or extrapolation). Ifthe spacing of the samples is fixed, then the values of L_(i) (t) dependonly on the position of time t relative to the sampling times used inthe interpolation. If this position is also fixed, then the interpolated(or extrapolated) value L_(n) (t,f) is a linear combination withconstant coefficients of the sampled values f(t_(i)) of the signal. Ifthe samples are equally spaced, then this distance will be called thescale of the Lagrangian polynomial. The scale can be shown to givefiltering features to polynomials, and thus to provide noise robustness.

It can be shown that the accuracy of polynomial approximations isexcellent in the unit subinterval in the center of the approximationinterval, and rapidly decays from the center of the approximation towardthe edge of the approximation interval, i.e., the interval spanned bysampling times used in the interpolation polynomial. This is illustratedin FIGS. 4 and 5 which compare a polynomial approximation of (sinπx)/(πx) with the function (sin πx)/(πx) on two different scales.

As shown in FIG. 4, around 0, the polynomial approximation of (sinπx)/(πx) is indistinguishable from the function (sin πx)/(πx) with anerror of 4×10⁻⁵. Between about -4 to +4, the error is indistinguishableat the resolution of the graph. However, the error grows to 10² ataround 10, and, as shown in FIG. 5, the error grows to 10¹¹ at around23, showing again the difference in nature between the polynomial andtrigonometric interpolations.

Therefore, the signal processing method and engine of the inventionreplace the original signal by piecewise highly smooth replicas,allowing the use of locally supported operators, while overall producingminimal distortion of the spectrum. For this reason, while unsuitablefor global methods, Lagrange's interpolation polynomial provides anexcellent starting point for obtaining a class of polynomials whichconvert global signal behavior information into local signal behaviorinformation.

This transformation of the global signal behavior information into thelocal signal behavior information also encodes spectral features of thewindowed sampled values of the signal used in the approximationpolynomial by the values of differential operators evaluated at pointsbelonging to the central unit interval of approximation.

This transformation of the global signal behavior information into thelocal signal behavior information is based on a Polynomial ApproximationTheorem which I posed and proved, and which is next presented.

F. Polynomial Approximation Theorem

As used herein, a π-band limited signal is a band limited signal whereinthe band limit is expressed in terms of radians rather than Hz.

Let f be a π-band limited signal such that ##EQU10## is finite, and let##EQU11## be the Lagrange interpolation polynomial with Nyquist rateinterpolation points f(i). Then, for sufficiently large n, ##EQU12##

Conversely, assume that a_(i) 's are real numbers so that ##EQU13## isfinite. Then ##EQU14## converges to a π-band limited signal f, such thatf(i)=a_(i). The rate of convergence depends directly on the distributionof the "energy" Σa_(i) ².

The above theorem is extensively used in the method of the invention toproduce weights, e.g. ##EQU15## or to define operators.

One important feature of the above theorem is that Lagrangianinterpolation polynomials of a band limited signal f(t) do notapproximate the signal f(t) "directly" but through a sequence of thewindowed signals ##EQU16##

This is a useful feature since exponential windows provide a goodrepresentation of the upper part of the spectrum of the entire signal byits finite windowed piece. This fact actually has a different meaningfor the method of the invention. The signal is so sharply windowed thatits Fourier transform is entirely dominated by the transform of thewindow. However, such windowing provides more faithful behavior of the"chromatic derivatives" (to be introduced below), which in turn producesa smoothing effect on the approximation of the Fourier transform of theresulting signal.

However, for smaller values of n, e.g. n less than about 25 to 65,depending on the desired accuracy, the window is too narrow, i.e. L^(n)_(i) (t) tend to be very small for all but a few values of i near thecenter of the approximation interval, and the polynomials L^(n) _(i) (t)are not sufficiently accurate approximations of functions ##EQU17##

Large values of n, e.g. n greater than about 30 to 60, provide a verysuitable window and an excellent approximation of functions λ^(n) _(i)(t) by L^(n) _(i) (t). But such large n produces too many polynomialsL^(n) _(i) (t) and the polynomials are of too large degree for practicalpurposes. An inspection of the values of these polynomials for -1<t<1shows that a substantial fraction of the polynomials do not havenumerically significant values, and thus can be omitted.

As used herein, the threshold of numerical significance is defined asfollows: A coefficient c multiplying a sampled value f(i) in anexpression is below the threshold of numerical significance if theproduct of c and the maximum value of the signal is below the resolutionof the system and on the level of or below the round-off error. Thethreshold of numerical significance is illustrated in FIG. 6 which isdescribed below.

The threshold of numerical significance here depends on the particularuse of approximation polynomials. In some instances interpolationpolynomials are used to interpolate from the Nyquist rate data. In suchcases n is relatively large (e.g. 32-64), so that the correspondingwindow is sufficiently wide. In such case, k is sufficiently large thatthe value of L^(n) _(k) (t) multiplied by the full scale of the inputsignal is below the resolution of the signal processing system, i.e.below the threshold of numerical significance. Such interpolations willbe nearly always applied to the values of differential operators andthus will be followed by or be a part of an integration procedureattenuating the high frequency part of the spectrum of the interpolationerror. In other applications, both n and k will be much smaller, sincethus obtained Nyquist rate data will be used only as a constraint for aleast square fit, and a much smaller number of sampled values will besufficient.

Thus, instead of using the polynomial ##EQU18## or ##EQU19## the methodof the invention uses ##EQU20## where k is substantially smaller than n.For example, in one embodiment, n=23, and k=8.

Further, inspecting the coefficients of L^(2n) _(i) (t) and L^(2n+1)_(i) (t), shows that the highest powers of t have small coefficients,and that only a small subset of all of these powers have numericallysignificant coefficients such that for -1<t<1 their impact can besignificant. Thus, after multiplication and regrouping of the samepowers of t, the highest powers of t in L^(2n) (t) and L^(2n+1) _(i) (t)can be dropped. In an embodiment in which n=23 and k=8, the polynomialis of degree 16.

In this way, k approximation polynomials, L^(2n+1) _(i) (t) and L^(2n)_(i) (t) (the tilde, "˜", indicating approximation), are obtained ofdegree m (<n). The exact values of k and m depend on the particularapplication, but are usually between 6 and 12 for k and between 12 and18 for m. The corresponding interpolation polynomials ##EQU21## aredenoted (here, using a bold L) by

    L.sup.2n+1 .sub.2k+1 (t,f), L.sup.2n.sub.k1,k2 (t,f)

or may be denoted by just L(t,f) if no confusion (in the superscriptsand subscripts) occurs, and is referred to herein as the basicapproximation polynomial. Note that while t must belong to a smallcentral sub-interval of the entire interpolation interval consisting of2n or 2n+1 points, k₁ can be taken slightly different from k₂ if needed.In the signal processing engine of the invention, this basicapproximation polynomial may be implemented in the form of a transversalfilter. This is illustrated below. The above polynomials are allcentered at 0. They can be shifted to obtain polynomials centered at anarbitrary sampling point s thus obtaining, for example ##EQU22##

However, polynomial approximations used in the method and engine of theinvention are unlike the wavelets used in the wavelet technique in thatthere is no single "mother wavelet." Each of the functions (sinπ(t-j))/(π(t-j)) must have a separate interpolation polynomial. This isdue to the fact that polynomial approximations are accurate only intheir central area, and so for different j and k, the approximationpolynomial for (sin π(t-j))/(π(t-j)) cannot be obtained by simply movingt to a new position within the approximation interval of theapproximation polynomial for (sin π(t-k))/(π(t-k)).

The described construction of approximation polynomials has thefollowing effects:

Let D be the interval of time containing exactly the sampling pointsappearing in the approximation polynomial L(t,f), including the zerovalues corresponding to L^(2n) _(i) (t) and L^(2n+1) _(i) (t) which weredropped (n≧|i|>k);

Let S be {i:|i|≧k};

D is called the interpolation domain and S is called the support of theapproximation polynomial;

Let C be the central subinterval of D of length equal to one or two unitintervals;

Then, the values of the polynomial approximation in the centralsubinterval C are obtained by:

(i) Setting f(i) to 0 at all sampling points outside the interval S, theinterval S having length 2k or 2k+1 (only these non-zero points will becounted and thus the corresponding approximation polynomial will be saidto have an even (2k) number of points or an odd number of points(2k+1)).

(ii) The values of the polynomial approximation L(t,f) in the interval Cobtained from the sampled values in S approximate a band limited signalwhich, outside D equals 0, and at any sampled values within D equals thewindowed sample values ##EQU23##

Thus, for an odd number of interpolation points (2k+1), ##EQU24## and,for an even number of interpolation points (2k), ##EQU25## and ##EQU26##

The effects of such windowing can be seen in the graph illustrated inFIG. 6. FIG. 6 is a graph of a signal f which is being sampled, and, inan interval D from sampled points -k to k, windowed by the factor##EQU27##

Outside the interval D, all sampled values are forced to 0. Within acentral interval C, the windowed values are represented by a Lagrangianapproximation polynomial.

The graphs of FIGS. 7A-7F represent the error of approximation of (sinπt)/(πt) by the Lagrangian polynomial L²×23+1 (t), i.e. L⁴⁷ ₀ (t):##EQU28##

The graphs of FIGS. 7A-7F show that Lagrangian approximations areextremely accurate in the interval (-1,1) with an error≦4×10⁻⁵, whichsharply drops to 6×10⁻⁴, i.e., by an order of magnitude, in the interval(-2,2). In the interval (-10,10), the error reaches a value of 3, while(sin πx)/(πx) at x=10.5 is less than 1/30, i.e., the error is more thanone hundred times larger than the value of the function approximated.Finally, on the edges of the interpolation interval, the errorapproaches 10¹¹. Thus, the difference between Lagrangian approximationsand trigonometric approximations is complementary. Polynomialapproximations fail globally and are excellent locally, whiletrigonometric approximations fail badly locally and are excellentglobally.

Thus, trigonometric interpolation of ##EQU29## "spreads" the Nyquistrate data provided by f(i) over long intervals, while polynomialapproximations ##EQU30## concentrate such data to the small centralsubinterval of the approximation interval, but there they provide a verysmooth and (for sufficiently large k) reasonably accurate approximationof the signal. This is also true of the first few derivatives of (sinπx)/(πx) and the corresponding derivatives of the polynomialapproximation.

IV. Signal Processing Based on Lagrangian Approximation Polynomials

The fact that Lagrangian polynomials with Nyquist rate equally spacedinterpolation points can approximate arbitrarily accurately any givenπ-band limited signal through a sequence of approximations obtained bywindowing, has several important consequences. In general, increasingthe number of interpolation points does not increase the accuracy of apolynomial approximation of an arbitrary function. Often only low degreepolynomials can be used, e.g. for prediction-correction methods forsolving differential equations.

The fact that increasing the number of interpolation points increasesthe accuracy makes it possible to design numerical algorithms forcalculus on band limited signals which can apply only to such signals,but which, for such signals, are extremely efficient.

A. Approximation Polynomials Represented by Transversal Filters

Linear operators applied to polynomial approximations may be representedby transversal filters in the following manner. Let A be a linearoperator which will be applied to a signal f. Then, the linear operatorA, applied to a polynomial approximation of the signal f, at a centralpoint 0 of the approximation, may be expressed as ##EQU31##

In accordance with the invention, an embodiment of a transversal filterwhich computes the output of a linear operator of a polynomialapproximation of a signal f, at a central point 0 of the approximation,may be constructed in a form illustrated in FIG. 8. In FIG. 8, anembodiment of a first general transversal filter 10 is formed by thecombination of delay elements 12, amplifiers 14, and adders 16. Thefirst general transversal filter 10 spans 2k+1 sampling times and 2ksampling intervals. The 2k+1 sampling times, corresponding to samplingtimes from t_(k) to t_(-k), where t_(k) represents the current or mostrecent sampling time, and t_(-k) represents the earliest sampling timein the sequence, i.e. 2k sampling times earlier, are represented by thejunctions 18 between the respective delay elements 12.

In descriptions of embodiments of transversal filters herein, areference to a general component will be by the general referencenumeral, e.g. 12, 14, 16, 18. In descriptions of embodiments oftransversal filters herein, a reference to a specific component will beby a specific reference numeral which identifies the respective generalcomponents, e.g. 12, 14, 16, 18, with a subscript identifying theposition of the particular component in the particular transversalfilter. For example, each junction 18 is allocated a specific referencenumeral corresponding to the sampling time represented by the junction18, the junction at t_(k) is identified by reference numeral 18_(k), andthe junction at t₀ is identified by reference numeral 18₀. Likewise,each amplifier 14 is allocated a reference numeral corresponding to thejunction 18, and hence sampling time, to which the amplifier's 14 inputis coupled.

For the purpose of identification only, a delay element 12 is allocateda reference numeral corresponding to the sampling time preceding (to theright of) the interval represented by the delay element, i.e. thejunction 18 following and coupled to the output of the respective delayelement 12. For example, the delay element 12 between sampling times t₁and t₀ is identified by reference numeral 12₀, and the delay element 12between sampling times t₀ and t₋₁ is identified by reference numeral12₋₁.

Likewise, adders coupled in the configuration illustrated in FIG. 8 willnormally be identified by a reference numeral corresponding to theamplifier 14, and hence the junction 18 representing the earliestsampling time (rightmost) coupled to the input of the respective adder16. Since there would generally be only one input at t_(k), theretypically will not be an adder 16 shown coupled only to the leftmostamplifier 14_(k). A reference to a component at a particular butunspecified position in the first general transversal filter 10 will bereference, e.g., to 12_(i), where i is within the limits (k to -k, ork-1 to -k) corresponding to the respective component.

In descriptions of embodiments of transversal filters implementing twoor more linear operators independently operating on polynomialapproximations of the same signal, e.g., in FIGS. 11 and 12 (describedbelow), a second set of amplifiers and adders will be identified by thegeneral reference numerals 24, 26, and so on for each additional set ofamplifiers and adders.

The interval of time represented by a delay element 12 will generally bean oversampled rate interval, but, if so indicated, may instead be aNyquist rate interval.

In order to simplify the FIGs. and the descriptions of the FIGs., all ofthe components are allocated reference numerals only in each of thefirst four exemplary embodiments of general transversal filtersillustrated in FIGS. 8, 9, 11 and 12. In subsequent FIGs., onlyrepresentative components, including components which are expresslyreferred to in the specification, are allocated reference numerals.Also, those components in the various embodiments of transversal filtersillustrated in the FIGs. are generally allocated the same set ofreference numerals as those shown in FIGS. 8, 9, 11 and 12, e.g. 12representing delay elements, 14 representing amplifiers, 16 representingadders, and 18 representing junctions between delay elements 12. In someinstances, these reference numerals may be shown with a prime, e.g. 16'representing a single adder substituted for a plurality of adders in aparticular transversal filter.

Referring again to FIG. 8, the 2k delay elements 12 are coupled insequence, with the most recent sample of the signal at t_(k), i.e.junction 18k, input to delay element 12_(k-1) and amplifier 14_(k), andpassed to succeeding delay elements 12 at the specified sampling rate.The 2k+1 amplifiers 14 have their inputs coupled to the respectivejunctions 18. Coupled to the input of a first adder 16_(k-1) are theoutputs of the two leftmost amplifiers 14_(k), 14_(k-1). Coupled to theinput of each succeeding adder 16_(i) is the output of the correspondingrespective amplifier 14_(i) and the preceding adder 16_(i+1).

The gain of each respective amplifier 14_(i) corresponds to acoefficient, c_(i), in the polynomial approximation above where, for thelinear operator A, each c_(i) may be calculated

    c.sub.i =A(L.sup.2n+1.sub.i)(0).

The output from the first general transversal filter 10 is the outputfrom the rightmost adder 16_(-k), and represents the polynomialapproximation of the operator A, applied to the signal f, at centralpoint 0 (corresponding to t₀).

The first general transversal filter 10 illustrated in FIG. 8 includes2k adders 16. While it may be necessary, or useful, in someapplications, to have access to interim sums, it is normally the casethat the only desired output is from the rightmost adder 16_(-k). Theconfiguration illustrated in the embodiment shown in FIG. 9 may then beused. FIG. 9 illustrates a second general transversal filter 20 in whichthe 2k adders 16 illustrated in the diagram of FIG. 8 are replaced by asingle adder 16'. All other components remain unchanged in theconfiguration illustrated in FIG. 9.

Since, in typical digital implementations, the entire transversal filtermay be implemented in a single microprocessor, the distinction betweenthe multiple adders 16 and the single adder 16' becomes animplementation detail. For example, FIG. 10 illustrates a flow chart ofan embodiment of a transversal filter routine 30 which is equivalent tothe first and second general transversal filters 10, 20 illustrated inFIGS. 8 and 9.

Referring to FIG. 10, the transversal filter routine 30 enters 32 andbegins processing in an initialization block 33. In the initializationblock 33, a sum S is initialized to 0, a temporary sample value a_(k+1)is assigned the value of the current sample of the signal, and an indexi is set equal to -k-1 (temporarily pointing to a fictitious junction18_(-k-1)). Then, in a processing block 34, i is incremented by 1, nowhaving the value -k (now pointing to the rightmost junction 18, junction18_(-k)), and the value of a_(-k) is assigned the value of a_(-k+1).This is equivalent to shifting the value of a sample through delayelement 12_(-k). The product of a_(-k) and c_(-k) (representingamplifier 14_(-k)) is then added to the sum S.

In a test block 35, the index i is compared with k (representingjunction 18k) to determine if the computation is complete (computationis complete when i=k). If the computation is complete, the transversalfilter routine exits 36. If the computation is not complete, processingcontinues through the loop until the computation is complete.

In the embodiment of the transversal filter routine 30 illustrated inFIG. 10, the processing moves from right to left along the transversalfilter. The processing could likewise proceed from left to right, takingcare to temporarily save the respective sample values so that they arenot overwritten and lost in the shifting process.

As will be seen below in the description of a transversal filterillustrated in FIG. 13, by arranging a diagram of a transversal filterwith multiple adders 16, by indicating a direction, or directions, offlow through the adders 16, and by showing the result output from aparticular adder, e.g. 16₀, helps to emphasize the point that the linearoperator is applied to the polynomial approximation of the signal at thecentral point 0.

Two or more linear operators independently operating on polynomialapproximations of the same signal may be represented by a third generaltransversal filter 40 arrangement as illustrated in FIG. 11. In FIG. 11,the third general transversal filter 40 includes a single set of delayelements 12, with two rows each of amplifiers 14, 24 and adders 16, 26.A first combination of amplifiers 14 and adders 16 coupled to the set ofdelay elements 12 forms a first transversal filter 42, and a secondcombination of amplifiers 24 and adders 26 coupled to the set of delayelements 12 forms a second transversal filter 44. The first row ofamplifiers 14 represents a set of coefficients corresponding to a firstlinear operator A applied to a polynomial approximation of the inputsignal, and the second row of amplifiers 24 represents a set ofcoefficients corresponding to a second linear operator B applied to apolynomial approximation of the input signal. The output of the firsttransversal filter 42, at adder 16_(-k) is [A(f)](t₀), and the output ofthe second transversal filter 44 at adder 26_(-k) is [B(f)](t₀).

The two respective rows of adders 16, 26 in the embodiment illustratedin FIG. 11 may each be represented by a single adder as illustrated inFIG. 9. This is illustrated in an embodiment of a fourth generaltransversal filter 50 in FIG. 12. In the fourth general transversalfilter 50, two rows of adders 16, 26 shown in the diagram of FIG. 11 areeach replaced by a single adder 16', 26'.

In both FIGS. 11 and 12, additional linear operators applied topolynomial approximations the input signal may be included in therespective arrangements by coupling additional rows of amplifiers 14 andadders 16 to the respective junctions 18.

FIG. 13 is a diagram of an embodiment of a fifth general transversalfilter 60 which may be used to implement a least square fit procedureapplied to a polynomial approximation of an oversampled signal. It willbe shown below, in the presentation of the Finite Local ε-base Theorem,that a transversal filter of the form of the fifth general transversalfilter 60 illustrated in FIG. 13 can be used to obtain a least squarefit for oversampled data contained in a small interval I, in which thesolution is a linear combination of Nyquist rate samples f(i), -n≦i≦n,and oversampled values f(τ_(j)) of the form ##EQU32##

In the fifth general transversal filter 60 illustrated in FIG. 13, theinput signal is being sampled (not illustrated), and input to thetransversal filter 60 at three times the Nyquist rate. In the embodimentillustrated in FIG. 13, the transversal filter includes 43 samplingtimes (from t₂₁ to t₋₂₁) and 42 sampling intervals corresponding to 42delay elements 12. In a "dense" interval corresponding to I, rangingfrom t₉ to t₋₉, the transversal filter 60 includes an amplifier 14 (andcorresponding adder 16) coupled to each junction 18 representing arespective sampling time. This dense interval will be shown below tocorrespond to the "central interval" of either a signal processing"micro-universe" (defined below) represented by the transversal filter,or a union of central intervals of a sequence of such micro-universes,i.e. a "complex" (which is defined below). However, outside the centralinterval, there is an amplifier 14 and adder 16 coupled to a junction 18between every third delay element 12 (i.e. every third samplinginterval). Thus, outside the central interval, the transversal filter 60applies the linear operator to the oversampled signal, but only at theNyquist rate, not at the oversampled rate according to the equation fora_(k) above. Also, in FIG. 13, emphasizing the point that the linearoperator is applied to the polynomial approximation of the signal at thecentral point 0 of the approximation, the arrows on the lines couplingthe adders 16 all point toward the adder 16₀ corresponding to t₀.

In the embodiment of a fifth general transversal filter 60 illustratedin FIG. 13, as shown in FIGS. 9 and 12, the adders 16 may be replaced bya single adder which has the oversampled rate inputs within the centralregion of the filter and the Nyquist rate inputs elsewhere.

The embodiments of transversal filters illustrated in FIGS. 8, 9, 11, 12and 13 are general forms of preferred embodiments of transversal filterswhich may be implemented in accordance with the invention to obtain theoutput of a linear operator applied to a polynomial approximation of asignal at a central point of the approximation. The form of the equationfor such polynomial approximations is shown above, as is the equationfor calculating the coefficients which are represented by the respectiveamplifiers in the transversal filters.

B. Differentiation

The fact that ##EQU33## with an error of the form ##EQU34## can be shownto imply that for m^(th) derivative ##EQU35## with an error of the form##EQU36##

Thus, an approximation of the derivatives of the signal at a givenpoint, can be obtained by differentiating the Lagrangian polynomials(L^(2n+1) _(i) (t)).sup.(m) and evaluating these formal derivatives atthe given point. This can be performed at the central point of theapproximation using the corresponding transversal filter for the m^(th)derivative. In general, for any approximation polynomial P_(j) (t), thecoefficients for differentiation, c_(i), are calculated as P.sup.(m)_(j)(0), where P.sup.(m)_(j) (t) are the m^(th) derivatives of theapproximation polynomials. For a transversal filter representing them^(th) derivative, the coefficients, and hence the gain of eachrespective amplifier, may be calculated as

    c.sub.i =(L.sup.2n+1.sub.i (0)).sup.(m).

FIG. 14 is a diagram of an embodiment of a general transversal filterfor the m^(th) derivative 70. The general transversal filter for them^(th) derivative 70 has the basic layout of the first generaltransversal filter 10 illustrated in FIG. 8. The coefficientsrepresented by the amplifiers 14 have values, e.g., for amplifier14_(n), C_(n) =(L^(2n+1) _(n) (0)).sup.(m), for amplifier 14₀, c₀=(L^(2n+1) ₀ (0)).sup.(m), and for amplifier 14_(-n), c_(-n) =(L^(2n+1)_(-n) (0)).sup.(m).

Problems of Differentiation in Signal Processing

It is known in numerical analysis that narrower spacing produces smallerapproximation error in differentiation but dangerously increases thesensitivity to the round-off error. However, band limited signals allowa fundamentally better and more accurate approach by altering theNyquist rate sampled values in order to ensure a better fit to theoversampled values. Thus, whenever possible, the signal processingmethod and engine of the invention differentiate only polynomialsobtained from the Nyquist rate Lagrange interpolation polynomials, withinterpolation values altered through a quadratic minimization procedureusing oversampled values. This solves the problem of differentiation forlow order derivatives (up to 2-4). However, higher order derivativessuffer from another, very serious problem. The higher order derivativesare, from the point of view of the spectral structure of the signal,non-informative, since their values are entirely determined by anextremely narrow highest band of the spectrum because the low harmonicsbecome negligibly present in the spectrum of the derivative.

FIGS. 15A-15D are graphs of |ω/π|^(2j) for j=1, 2, 3, 4, 5, 6, 15 and 16appearing as a multiplicative factor in the energy expression, butnormalized so that the maximum value of |ω/π^(2j) is 1: ##EQU37## of thej^(th) derivative ##EQU38## of f(t).

However, this problem is not in the differentiation per se, but in thefact that derivatives are an extremely inconvenient base for the algebraof linear differential operators. For this purpose, the inventionreplaces the base consisting of the set {f.sup.(j) (t): j.di-electcons.N} by the following recursively defined sequence of differentialoperators:

D.sup.(0) (f)≡f(t),

D.sup.(1) (f)=f'(t)/π,

D.sup.(2) (t)=2f.sup.(2) (t)/π² +f(t),

D.sup.(3) (t)=4f.sup.(3) (t)/π³ +3f'(t)/π,

D.sup.(4) (t)=8f.sup.(4) (t)/π⁴ +8f.sup.(2) (t)/π² +f(t),

D.sup.(5) (t)=16f.sup.(5) (t)/π⁵ +20f.sup.(3) (t)/π³ +5f'(t)/π, and

D.sup.(6) (f)=32f.sup.(6) (t)/π⁶ +48f.sup.(4) (t)/π⁴ +18f.sup.(2) (t)/π²+f(t).

In general, the recursion formula is:

D.sup.(0) (f)=f(t),

D.sup.(1) (f)=1/πf'(t), and for n≧1,

D.sup.(n+1) (f)=2/π[D.sup.(n) (f)]'+D.sup.(n-1) (f).

A straightforward calculation shows the following property of the aboveoperators:

If ##EQU39## then ##EQU40## where T_(n) (x) is the n^(th) ChebychevPolynomial in the variable x, i.e. for

T₀ (x)=1,

T₁ (x)=x, and

T_(n+1) (x)=(2x)T_(n) (x)-T_(n-1) (x).

The significance of this is that T_(n) (ω/π)≦1 for all -π<ω<π, and thatT_(n) (ω/π)H(ω) encodes the spectrum of the signal very faithfully bycomb-filtering the Fourier Transform H(ω), giving more "detail" to thehigh end of the spectrum. This is obvious from the graphs of the factor|T_(n) (ω/π)|² appearing in the energy expression ##EQU41##

FIG. 16A is a graph of |T₀ (ω/π)|² and |T₁ (ω/π)|². FIG. 16B is a graphof |T₂ (ω/π)|² and |T₃ (ω/π)|². FIG. 16C is a graph of |T₄ (ω/π)|² and|T₅ (ω/π)|². FIG. 16D is a graph of |T₁₆ (ω/π)|². When this basis iscompared with regular derivatives, |f.sup.(16) (ω)| for ω=π has anextremely large value, π¹⁶, while for small ω it is essentially 0, e.g.for ω=1/(2π), |f.sup.(16) (ω)|<1/2¹⁶.

Because the differential operators D.sup.(n) of the invention encode thespectral features of the signal, they are referred to herein as"chromatic derivatives of order n."

There are other orthogonal families of polynomials having similarproperties, e.g. Legendre polynomials. However, Chebyshev polynomialshave an important property. They provide interpolations of continuousfunctions with a minimal pointwise ripple. This minimizes the maximalamplitude of spurious harmonics in the spectrum of the approximationerror.

The signal processing method and engine of the invention use extensivelythe fact that polynomials are smooth and that chromatic derivativesevaluated directly by applying a single linear combination withoutcalculating the derivatives separately encode a lot of information onthe spectral content of the signal. Thus, e.g., D.sup.(4) (f)(τ) may beobtained by a transversal filter with coefficients computed directly byevaluating

    ((8 P.sub.i.sup.(4) (t))/π.sup.4)+((8 P.sub.i.sup.(2) (t))/π.sup.2)+P.sub.i (t)

using analytic differentiation, combining the resulting polynomials,dividing the coefficients by π⁴ and only then substituting t=τ. Thisprevents the accumulation of round-off error.

Chromatic derivatives are the basis for the Taylorian paradigm. Nyquistrate data is collected by polynomial approximations, additionaloversampled data is added by enhancing the Nyquist rate data through aleast square fit procedure, and then spectral content of the signal,with emphasis on the high end of the spectrum is encoded into aTaylorian paradigm using chromatic derivatives. Essentially, suchdifferential operators account for the predictive features of theinvention. They also bridge the global-local behavior of band-limitedsignals by encoding the spectral (thus global) features of the signal inthe most local possible way, by the values of the chromatic derivatives.

C. Integration

As described above in the case of differentiation ##EQU42## implies##EQU43##

Whenever the interval [a, b] is of length at most one unit intervallong, with the error given by ##EQU44## In all relevant applications itis determined that these approximations are of sufficient accuracy.

Given Nyquist rate sampling points t₀ and t₁ several Nyquist rateintervals apart, and an approximation polynomial P(x,f) defined as alinear combination of polynomials P_(i) (x) multiplied by the sampledvalues of the signal f(t), then the integral ##EQU45## is obtained byrecursion from the locally supported integral operators ##EQU46## asfollows: ##EQU47## This is accomplished by a recursive infinite impulseresponse (IIR) system as illustrated in FIG. 17.

In FIG. 17, an IIR system 70 includes a transversal filter 76 coupled tothe input of an accumulator 72. The output of the accumulator is thepolynomial approximation of the integral above. The transversal filter76 produces an approximation of ##EQU48## The accumulator 72 sums theseintegrals to obtain ##EQU49## for some fixed starting point t₀corresponding to the reset time of the accumulator.

Clearly the earliest sample used in the above calculation is t_(0-n)=t_(-n) ; t₀ is the reset instant of the accumulator A. The values c_(j)of c_(-n) to c_(n) are equal to ##EQU50## which is explicitly calculatedfrom the definition of P_(j) (t) analytically (e.g. using Mathematica).

Integrals of the form ##EQU51## are intrinsically unstable, because IIRoperators are sensitive to low frequencies. A variant of the IIR system83 in FIG. 17 without a reset would calculate such integrals, but theaccumulator 82 can easily overflow unless appropriate measures aretaken. Integration can be made "leaky" if, for example, an accumulatorfor S_(i+1) :=S_(i) +x_(i+1) is replaced by S_(i+1) :=qS_(i) +x_(i+1)where q<1 and q≈1, e.g. q=0.999. (As used herein, :=is the assignmentoperator, commonly also represented by the left arrow ←.) This clearlyintroduces some distortion on the low end of the spectrum, but allintegration procedures over long intervals suffer from such "drifts". Ifsuch integration is followed by a differentiation as in Σ-Δ filteringprocedures, then the impact of the drift is relatively small. However, amore accurate solution applicable to Σ-Δ procedures, is described below.

D. Σ-Δ Procedures

If the signal is given in an analog format or on an oversampled scale,then either an analog integration can first be used, or an integrationusing summation of integrals over the unit length intervals obtainedfrom the appropriate polynomial approximations on an oversampled scale,can be used as described above. If the signal is very substantiallyoversampled, then a summation of the oversampled values can be dividedby the length of the oversampling interval to produce a goodapproximation of the integral ##EQU52##

The value I(t) can be then either sampled at the Nyquist rate scale (ifthe signal is given in the analog format) or, if the value of the aboveintegral is obtained by polynomial approximations and summation, thesevalues can be decimated to obtain values of this integral at samplingpoints spaced at the Nyquist rate unit interval. Applyingdifferentiation using Lagrangian polynomials produces a low passfiltering on the input signal which has been integrated. Thus, this is aΣ-Δ (sigma delta) filtering procedure. If the signal is oversampled at arate of only 4 times the Nyquist rate, then a filter with thecharacteristics illustrated in FIG. 18 is obtained. In a preferredembodiment, this Σ-Δ procedure may be integrated with a least square fitapproach designed for the case of signals with a very large level of outof band noise. The least square fit approach is provided by atransversal filter of the form of the fifth general transversal filter60 illustrated in FIG. 13, and described below in the presentation ofthe Finite Local ε-base Theorem. This theorem produces a filter with anextremely sharp cut off and low out of band ripple.

This procedure may be implemented by a filter of the structure shown inFIG. 19. Filter F1 is an integration transversal filter which produces##EQU53##

The accumulator provides ##EQU54## calculated on the oversampled scale.Filter F2 differentiates this integral on the Nyquist rate (cut-offfrequency) scale.

In order to avoid overflow of the accumulator the fact thatdifferentiation uses only finitely many values may be used, and the factthat the value of the derivative does not change if the same number issubtracted from all of the samples, i.e. (f(t)+constant)'=f'(t). Thus,the accumulator A can be replaced by two accumulators as shown in FIG.20.

While integrator A is in use, the accumulator of the integrator B isreset. Integrator A is used until the values from integrator B reach theend of the delay chain b. At this instant, the switch S connects to theoutput of the transversal filter B, which has different values from thevalues on the filter A, but the same differences between these values.This implies that the value of the derivative calculated by A and by Bare the same. However, now integrator A can be reset and the cyclerepeats. In this manner, the integrators do not overflow, because theyare reset frequently.

Such Σ-Δ procedures have no phase shifts (but possibly only delays. Ifthe predictive complex [to be introduced later] is used, no delay occursat the expense of larger ripple). Note the importance of the fact thatin such Σ-Δ procedures, integration and differentiation are on twodifferent scales. They can be implemented using charge coupled device(CCD) technology.

E. Solving Linear Differential Equation Using Lagrangian PolynomialApproximations

In order to solve linear differential operators it is enough to solvesuch equations of the first order, since any other equation can bereduced to a system of linear differential equations. The methoddescribed below entirely depends on the fact that increasing the numberof the Nyquist rate interpolation points improves the accuracy of theapproximation. Thus, to solve linear differential equations, a lowdegree polynomial is not needed for a predictor--corrector method.Instead, if in the equation

    f(t)+af'(t)=g(t)

if both f(t) and g(t) are given at the Nyquist rate samples, one canapproximate g(t) by a polynomial of relatively large degree, for example17 or above. Then the above equation implies

    f(t)=g(t)a.sup.-1 -f(t)a.sup.-1,

and by differentiating both sides results in a sequence of the form

    f.sup.(2) (t)=g'(t)a.sup.-1 -g(t)a.sup.-2 +f(t)a.sup.-2,

    f.sup.(3) (t)=g.sup.(2) (t)a.sup.-1 -g'(t)a.sup.-2 +g(t)a.sup.-3 -f(t)a.sup.-3,

therefore

    f.sup.(m) (t)=g.sup.(m-1) (t)a.sup.-1 g.sup.(m-2) (t)a.sup.-2 + . . . +(-1).sup.i g.sup.(m-i) (t)a.sup.-i . . . -(-1).sup.m g(t)a.sup.-m +(-1).sup.m f(t)a.sup.-m.

Thus, Taylor's formula implies that ##EQU55##

Let P(t,g) be an interpolation polynomial for g, and assume that f(i)has already been obtained. In the unit interval (i, i+1) this polynomialapproximates g with a high accuracy. Thus, in order to obtain thesolution in the same interval, Taylor's formula can be used with thederivatives of g(t) replaced by the derivatives of the approximatingpolynomial P(t,g): ##EQU56##

In order to evaluate the above expression the polynomial P(t,g) isrepresented as a sum of basic polynomials multiplied by the sampledvalues of the signal: ##EQU57## and differentiation is performed on eachof these polynomials separately. The coefficients are then multiplied bythe factors containing the parameter a, and thus producing values ofconstants (depending on a) which are then used to form the appropriatelinear combination of the sampled values. The reason for performing theprocedure in this order is to reduce the round-off error, because thecoefficients of higher order derivatives grow in size with the order ofdifferentiation.

Decreasing accuracy of approximations of higher order derivatives ofg(t) by the derivatives of P(t,g) is offset by the fact that higherorder derivatives are divided by factorials of large numbers. Thisreflects the fact that an approximation of the solution f(t) is soughtwithin a small, unit length interval, and correctness of the methodrelies again on the fact that the solution f(x) can be approximated by aLagrangian polynomial and that Taylor's formula for polynomials isexact. In fact, on such small intervals, the truncated Taylor's form ofa Nyquist rate spaced interpolation polynomial accurately represents thehigher degree polynomial. Thus, the number m above can be smaller thanthe degree of the polynomial P(t,g). Also, the error of polynomialapproximations which is in the high part of the spectrum is attenuatedwhen an integration procedure is applied.

A significant improvement of the above method consists in expressingTaylor's formula using the chromatic derivatives. Not only are thecoefficients thus obtained much smaller than the correspondingcoefficients for the "standard" derivatives, but then the increaseddegree of approximation by Taylor's polynomial provides a better matchof the spectral features of the solution and less out of band noise.

If the signal is oversampled, the procedure described in the processingpart of the engine is capable of performing both differentiation andintegration with exceedingly high accuracy surpassing the harmonicmethods by orders of magnitude.

V. Basic Signal Processing Local Universes

Signal processing in accordance with the invention is performed in smalloverlapping "local micro-universes," also referred to herein as "localuniverses," each local universe associated with a Nyquist rate samplingpoint and containing a relatively small number of samples of the signal.The set of the corresponding sampling points for these available samplesin a local universe is called the "support" of this local universe. Suchlocal universes have their "local" versions of the signal, and signalprocessing operators act using only data available in the correspondinglocal universes. However, by external conditions relating several suchlocal universes, the "local" versions of the signal and the actions ofthe signal processing operators are made consistent with each other. Inthis way, on the global scale, statistical laws of error distributioninsure low error in the global features of the signal in the frequencydomain.

A Finite Local ε-base Theorem, which I posed and proved, is presentednext. This theorem provides the justification why such processing ispossible. While functions (sin π(x-i))/(π(x-i)) form an orthonormalsystem, and are thus linearly independent, the Finite Local ε-baseTheorem shows that over a short interval (-1, 1), and for any givensmall number ε, finitely many of such functions are sufficient torepresent the signal over the interval (-1, 1) with pointwise errorsmaller than (ε×A_(max)), where A_(max) is the maximal amplitude of thesignal.

A. Finite Local ε-base Theorem

For every ε>0, there exists n and numbers a₁, . . . a_(n) <1 such that##EQU58## for all t such that -1≦t≦1.

The above theorem implies that given sufficiently large n, one can alterthe values of Nyquist rate points f(-n)^(i), . . . , f(0), . . . , f(n)into a_(-n), . . . , a₀, . . . , a_(n) such that in the interval (-1,1)the signal f*(t), defined such that f*(m)=0 for |m|>n and f*(i)=a_(i)for i<|n|, is an ε fit, i.e. a close fit, of f on the interval (-1,1).

It is possible to show that the energy of the difference ##EQU59##needed to accomplish the above local fit depends on the total energy ofthe signal and its distribution on the samples in the interval (-n, n)versus the samples outside this interval.

The main application of the above theorem is for obtaining a leastsquare fit for the oversampled data contained in a small interval I,using a Lagrangian interpolation polynomial with the interpolationvalues which are variations of the Nyquist rate sampled values of thesignal. In such applications, instead of trying to accomplish the bestpossible fit, a quadratic optimization is used to find a "compromise"between minimization of each of the following two magnitudes:

(1) the "energy" of the difference between the signal and its alteredvalues on the interpolation points of the polynomial approximation, bothwindowed by the windowing function corresponding to the polynomialapproximation ##EQU60## (2) the deviation of the interpolated valuesfrom the real oversampled values of the signal in the interval I:##EQU61## Thus, a weighted sum of the form Y₁ +vY₂ is minimized with alarge weight v heavily emphasizing the second term. This allows that avery significant part of the energy of the signal outside theinterpolation interval be taken into account.

Minimization of the above quadratic expression, Y₁ +vY₂, is done in theusual way. Partial derivatives of Y₁ +vY₂ with respect to each ofa_(-n), . . . , a₀, . . . , a_(n) are found and set equal to 0. Thisresults in a system of linear equations in variables a_(-n), . . . , a₀,. . . , a_(n). The solution of this system is a linear combination ofNyquist rate samples f(i), -n≦i≦n, and oversampled values f(τ_(j)) ofthe form ##EQU62## where i ranges over the Nyquist rate sampling pointsfrom -n to n, and τ ranges over the oversampling rate points, includingthe Nyquist rate points, within I. Thus each a_(k) for -n≦k≦n is theoutput of a transversal filter implementing the above linearcombination.

B. k-Monad

A k-monad M (also referred to herein as a "monad", k referring to thenumber of Nyquist rate sampling points in the support of the monad)consists of

i. An interval of time D_(M), beginning and ending with Nyquist ratesampling points, the interval containing either 2n or 2n+1 Nyquist ratesampling points, which interval of time is called the domain of themonad M;

ii. An interval of time S_(M), beginning and ending with Nyquist ratesampling points, the interval containing either 2k or 2k+1 Nyquist ratesampling points for some k<n, which interval of time is called thesupport of the monad M;

iii. A collection of real numbers, a_(i), each associated with thesampling point i belonging to S_(M). These numbers can be, but are notnecessarily the sampled values of a signal. They can also be obtainedthrough a computational procedure, e.g. a least square fit. For allsampling points in D_(M) which are not in S_(M), it is assumed that the"local" value of the signal at these sampling points is equal to zero(the corresponding a, are considered all equal to 0 and thus they arenot introduced at all.)

iv. An interval of time C_(M) of length one to two Nyquist rate unitintervals, either centered about the central point of the domain D_(M)(if D_(M) has an odd number of points) or centered between the twocentral points if D_(M) has an even number of points); and

V. An interpolation polynomial P(t, a), of the form ##EQU63## (bold asignifying a vector quantity).

Thus, the basic polynomial approximation can be seen as a monad witha_(i) =f(i), with the interpolation polynomial evaluated atinterpolation points belonging to C_(M). The values of a_(i) are calledthe output of the monad.

Monads can be used to capture local signal behavior if the signal is nottoo noisy and the oversampled rate is sufficiently high. However, amonad will typically be used to build more complex structures. Thus,monads may be viewed as elementary local universes of the signalprocessing method and engine of the invention.

Polynomial approximations naturally induce a windowing of the signal.Thus, a monad can be seen as containing an internal version of thesignal f(t) with windowed samples ##EQU64## This follows from thePolynomial Approximation Theorem, with only the part of the signal inthe central interval C being explicitly described by the correspondingpolynomial approximation. The domain of the monad is "the local versionof the entire time", while the support of a monad is the interval intime where the entire energy of the signal comes from, since the energyis equal to the sum of the squares of the Nyquist rate sampled values,and the support contains all samples which are not equal to 0. Noticethat the signal itself is not equal to 0 at all points between thesamples, but decays rapidly due to exponential windowing. In the centralinterval C the "internal" version of a signal g(i)≡a_(i) is given by thevalues of the polynomial approximation ##EQU65## This has the effect ofwindowing the signal g(i) by an exponential window ##EQU66## whichimplies that ##EQU67## because it is assumed that for all i which areoutside the support S_(M), a_(i) =0.

FIG. 21 is a graph which illustrates the characteristics of a monad.

C. p-Simplex

A p-simplex (also referred to herein as a "simplex", p referring to thenumber of Nyquist rate sampling intervals in the central interval) is acollection of monads parameterized by the same set of parameters a_(i),and with a piece-wise polynomial approximation defined as a weightedcombination of approximation polynomials of individual monads, withpolynomial weights. Thus, a simplex consists of:

i. A sequence of monads, M₁, . . . , M_(k) with domains D₁, . . . ,D_(k), supports S₁, . . . , S_(k), and central regions C₁, . . . ,C_(k). The domain, support or central region of a monad can partlyoverlap or be a subset of the domain, support or central region ofanother monad;

ii. An interval of time D_(S) beginning and ending with Nyquist ratesampling points, such that D_(S) =D₁ ∪ . . . ∪ D_(k) ;

iii. An interval of time S_(S) beginning and ending with Nyquist ratesampling points, such that S_(S) =S₁ ∪ . . . ∪ S_(k) ;

iv. A sequence a of real numbers, a={a_(i), i .di-elect cons. S_(S) },each a_(i) associated with a Nyquist rate sampling point i .di-electcons. S_(S) ; Thus, each monad M_(j) uses the same value a_(i) for i.di-elect cons. M_(j) ;

v. iv. A central interval C_(S) containing p Nyquist rate samplingintervals, with p usually equal to 2 or 3. The centers of C_(S) andD_(S) need not coincide, but the center of D_(S) must belong to thecenter of C_(S), and C_(S) =C₁ ∪ . . . ∪ C_(k) ; and

vi. A piece-wise polynomial approximation Q(t,a), for a={a_(i), i.di-elect cons. S_(S) }, defined as follows:

Let P₁ (t,a₁), . . . , P_(k) (t,a_(k)) be polynomial approximationsassociated with monads M₁, . . . , M_(k), where a_(j) ={a_(i), i.di-elect cons. S_(j) }. Let (t_(s), t_(s+1)) be a Nyquist rate unitsubinterval of C_(S).

Then there are polynomials R^(S) ₁ (t), . . . , R^(S) _(k) (t) such that

    R.sup.S.sub.1 (t)+ . . . +R.sup.S.sub.k (t)=1              (condition 1),

and such that

    Q(t,a)=Q.sub.S (t,a)=R.sup.S.sub.1 (t)P(t,a)+ . . . +R.sup.S.sub.k (t)P.sub.k (t,a)                                          (condition 2),

where some R^(S) _(i) can be 0 polynomials and are in fact 0 polynomialswhenever (t_(s), t_(s+1)) is not a sub-interval of the central intervalC_(i) of the corresponding monad M_(i). At the end points of suchintervals, i.e. at the Nyquist rate sampling points in C_(S).

    Q(t.sub.S,a)=(Q.sub.S (t,a)+Q.sub.S-1 (t,a))/2 .

Clearly, this, together with conditions 1 and 2 implies

    Q(t.sub.S,a)=a.sub.S.

A primary example of a simplex is called a "link", denoted by Λ. FIG. 22is a graph which illustrates a link. A link consists of three monadsM_(L), M_(R) and M_(C), which are the left, right and central monads,respectively. The domain D.sub.Λ of Λ contains 2n+1 Nyquist ratesampling points t_(-n), . . . , t₀, . . . , t_(n). The domain D_(C) ofthe central monad is the entire domain D.sub.Λ of the link Λ. The domainof the left monad contains an even number of points {t_(-n), . . . ,t_(n-1) } while the domain of the right monad contains points {t_(-n+1),. . . , t_(n) }. Thus, the domain of M_(L) is missing only theright-most point t_(-n) of D.sub.Λ, and the domain of M_(R) is missingonly the leftmost point t_(-n). The support S.sub.Λ of Λ contains 2k+1Nyquist rate sampling points t_(-k), . . . t₀, . . . , t_(k), and allthree monads have the same support. The central region of Λ, C.sub.Λ, isthe interval (t₋₁, t₁) and has the following piece-wise polynomialapproximation associated with it:

For t .di-elect cons. (t₀, t₁)

    Q(t,a)=Q.sub.1 (t,a)=(1-t)P.sub.C (t,a)+tP*.sub.R (t,a).

Thus, R¹ ₁ (t)=1-t and R¹ ₂ (t)=t, P_(C) (t,a) is the Lagrangianpolynomial associated with the central monad (odd number ofinterpolation points) and P*_(R) (t,a) is obtained from the Lagrangianpolynomial associated with the right monad P_(R) (t,a) (even number ofinterpolation points), with the window adjusted to coincide with thewindow of the central monad.

Similarly, for t .di-elect cons. (t₋₁,t₀)

    Q(t,a)=Q.sub.-1 (t,a)=(1+t)P.sub.C (t,a)-tP*.sub.L (t,a)

For the case of Lagrangian interpolation polynomials this results in##EQU68##

Another convenient definition for Q(t,a) which produces slightly moreout of band interpolation noise is given by a single polynomial

    Q(t,a)=(1-t)/2P*.sub.i (t,a)+(1+t)/2P*.sub.R (t,a).

Note that this does not reduce the simplex into a monad because thedomains of M_(L) and M_(R) are different, i.e. P_(i) (t,a) and P_(R)(t,a) are equal to 0 over two different sets of sampling points. Bothdefinitions are extremely accurate. The following illustrates theaccuracy of the first piece-wise polynomial approximation given abovefor the signal f(t)=(sin πt)/(πt). M_(j) denotes the maximal value ofthe j^(th) derivative of the signal, i.e. max {f.sup.(j) (t)}.

    |Q(t,a)-f(t)|≦4×10.sup.-5 ×M.sub.0,

    |Q'(t,a)-f'(t)|≦1×10.sup.-4 ×M.sub.1,

    |Q.sup.(2) (t,a)-f.sup.(2) (t)|≦5×10.sup.-4 ×M.sub.2,

    |Q.sup.(3) (t,a)-f.sup.(3) (t)|≦1.5×10.sup.-3 M.sub.3,

    |Q.sup.(4) (t,a)-f.sup.(4) (t)|≦2.5×10.sup.-3 ×M.sub.4,

    |Q.sup.(5) (t,a)-f.sup.(5) (t)|≦8×10.sup.-3 ×M.sub.5,

and

    |Q.sup.(6) (t,a)-f.sup.(6) (t)|≦1.3×10.sup.-2 ×M.sub.6.

Approximations of (sin π(t-j))/(π(t-j)), j>0, due to windowing, haveeven smaller errors.

A simplex can be used to capture local signal behavior in a more preciseway than using a monad. However, as monads may be considered elementaryparticles, a simplex may be viewed as an "atom" in the main sequence ofsignal processing micro-universes, i.e. in an m-complex, defined below.

D. m-Complex, Sequences of Local Universes

An m-complex, X, (also referred to herein as a "complex", m referring tothe number of links) is a sequence of m links Λ_(i) with domains D_(i),supports S_(i) and central regions C_(i) such that:

1. For any two adjacent links the right monad of Λ_(i) and the leftmonad of Λ_(i+1) share the same domain;

2. The supports S_(i) and S_(i+1) of each two consecutive links shareall points except the leftmost point of S_(i) and the rightmost point ofS_(i+1) ; and

3. The central regions C_(i) and C_(i+1) overlap on one unit interval.

4. Each link Λ_(i) has its own parameterization {a^(i) _(j) |j.di-electcons. D_(i) }.

The condition 4 explains why a complex is not a local universe but asequence of local universes. Each link has its own internal version ofthe signal, and these versions can be different. However, a complex willhave conditions constraining to what degree these versions can bedifferent.

The focal point F_(X) of a complex X consisting of m links is thecentral point of any link Λ_(i). The choice of the focal point dependson the particular application of the complex. The most importantcomplexes are the complexes with an odd number of links and with thefocal point F_(X) coinciding with the central point of the domain of thecentral link, as well as a complex with the focal point coinciding withthe central point of the leftmost link or of the rightmost link.

The graph of FIG. 24, below, illustrates a complex with 9 links and withthe focal point F_(C) coinciding with the central point of the supportof the central link.

An independent signal processing is performed within each link, withdata available within each respective link support and accordinglywindowed. However, an external condition is imposed which insures thatadjacent links are in agreement as to what are the signals within theshared part of their corresponding supports. Also, wherever the sampledvalues are available, the internal versions of these values for eachlink are ensured to be sufficiently close to the actual sampled values.The deviation of all such parameters is weighted and the sum of suchweighted deviation is minimized. This procedure is described in detailbelow in Section VIII pertaining to the derivation of local signalbehavior processing parameters.

The output of a complex can be either

1. the sequence of values {D.sup.(0) (f(F_(X))), . . . , D.sup.(m)f(F_(X)))} where the number of chromatic derivatives evaluated dependson application, but is typically between 4-8,

or

2. The sequence {a^(c) _(j), |j.di-elect cons. D_(c) } where Λ_(C) isthe central link containing the focal point.

The choice of which data is to be output depends on the nature of thesubsequent processing of the output data.

The m-complex is the core of the basic engine for LSB signal processingin accordance with the invention. The complex allows simple and uniformdesigns of many algorithms. This will be clear from the applications ofcomplexes for the purpose of derivation of local signal behaviorprocessing parameters.

VI. Local Signal Behavior Parameters

The local signal behavior parameters for the class of π-band limitedsignals is a set of linear operators {δ₀ (f), . . . , δ_(k) (f)} whichhas the following properties for every set of Nyquist rate samplingpoints t_(i), i an integer.

1. For every ε, ε there exists a sequence ε₀, . . . , ε_(n) such thatfor arbitrary two π-band limited signals f and g the following holds. Iffor every sampling point t_(i) and for all j≦k |(δ_(j) (f)(t_(i))-(δ_(j)(g)(t_(i))|<ε_(j), then:

a. |f(t)-g(t)|<ε, for all t; and

b. Fourier transforms H_(f) and H_(g) of f and g respectively, satisfy

    |H.sub.f (ω)-H.sub.g (ω)|<ε for all -π<ω<π.

2. The values (δ₀ (f)(t_(i)), . . . , (δ_(n) (f)(t_(i)) can all beobtained from the samples of f(t) taken at sampling points belonging tothe interval (t_(i-1), t_(i+1)) at an oversampling rate which depends on

a. The presence of noise in these samples (including the quantizationnoise);

b. The level of accuracy required, i.e. on the magnitudes of ε and ε;

and

c. The delay allowed.

In practice an interval containing oversampled values of the signal willbe used, lasting 2-20 Nyquist rate unit intervals.

A primary example of local signal behavior parameters are the chromaticderivatives. They are clearly linear operators. Since chromaticderivatives up to the order k allow an approximation of the standardderivatives up to the same order, Taylor's formula ensures the propertyi-a., above. It can be shown that the fact that Chebychev polynomialsapproximate any continuous function with minimal pointwise error (andmost often with a minimal number of terms) ensures property i.b. above.

VII. Interpolation Polynomials in General

The invention contemplates that polynomial approximations, or piecewisepolynomial approximations, other than Lagrangian polynomialapproximations may be used. In accordance with the signal processingmethod and engine of the invention, several types of polynomialapproximations may be used to represent the behavior of a signal on thelocal level. For longer intervals, polynomial approximations become veryunsuitable since the degree of the polynomial needed to approximate thesignal accurately grows very rapidly (see the error estimate forTaylor's polynomials). On the local level, polynomial approximations aremost suitable for the following reasons:

1. Polynomial approximations are computationally the simplest possibleapproximations; and

2. Polynomial approximations are stable to differentiate, by evaluatingthe actual formal derivatives. The degree of smoothness is limited onlyby their degree.

However, not all polynomial approximations are equally suitable forrepresenting the local signal behavior. The criteria which should besatisfied by polynomial approximation suitable for use in accordancewith the invention are:

1. Both the approximation polynomials and their derivatives ofsufficiently high order must have a form with small coefficients inorder to be sufficiently stable with respect to the round off error.

2. The polynomials must be represented in a form directly parameterizedby the sampled values, i.e. as a sum ##EQU69## where P_(i) (t) are somepolynomial factors, and f(i) are the sampled values of the signal.

3. The polynomials must be "frequency selective," i.e. for τ in theinterval (t-1, t+1) P(τ,f)≈g(τ) where g is a band limited signal.Lagrangian interpolation polynomials have this property due to theirnatural notion of scale, which is the distance between their equallyspaced interpolation points. In fact, this distance defines the Nyquistfrequency for their band-limit.

4. The band limited signal g(t) above, such that P(τ,f)≈g(τ), must be asuitable windowed version of f(τ). For the Lagrangian polynomial with2n+1 interpolation points, this is the windowed signal f(t) with thefollowing window: ##EQU70## This property ensures an adequatepreservation of the spectral features of the signal.

Thus the linear operators defined using Lagrangian polynomials will alsohave rational coefficients in their corresponding matrices. Usingsoftware for symbolic manipulations, such operators may be composedwithout any round-off error accumulation. The rounding off andmultiplication with the sampled values can be done in the final stage ofcomputation only.

This is important because relatively unstable operators can be composedas long as the entire composition is a stable operator. Examples arepredictions which are an intermediate stage for evaluating the values ofdifferential operators at the very end of the data stream. Predictionsof the Nyquist rate future values of the signal rapidly deteriorates inaccuracy, but mostly by a low-frequency off-set. However, the fact thatthe coefficients of the Lagrangian derivatives rapidly decrease towardthe end-points of the interpolation interval very significantly reducesthe impact of the fact that the predicted values become increasinglyinaccurate.

The "globalizing" operators which combine past Nyquist rate data andpast oversampled data in order to obtain Nyquist rate predicted data areintrinsically unstable, i.e. they produce decreasingly accurate datatoward the end of the prediction interval, due to the windowing featuresof polynomial approximations. However, when such "globalizing" operatorsare composed with highly "localizing" operators like differentiation,the resulting composition is a stable "local+global→local" operator.

In general, the result of applying a composition of a sequence ofoperators to the signal should always be obtained by operating on thecoefficients obtained from the corresponding polynomial approximationswhich are used to define these operators, and only then the resultingcoefficients will be multiplied with the sampled data. This maintainsthe size of the coefficients as small as possible, prevents theround-off error accumulation, and minimizes the total number ofmultiplications required during the actual signal processing.

VIII. Derivation of Local Signal Behavior Processing Parameters

The invention employs oversampling to enable capturing the localbehavior of a signal. As defined herein, oversampling means sampling atn times the band limit of the signal, where n is greater than 2 (withn=2 commonly referred to as the Nyquist rate), In accordance with thesignal processing method and engine of the invention, the particularembodiment of the invention used, and the parameters captured todescribe the local signal behavior may depend on several factorsincluding the degree of out of band noise in the signal, the degree ofoversampling to be employed and the maximal allowed delay.

Example: The delay allowed is at least 10 Nyquist rate unit intervals.

A complex is formed having 3 to 7 links depending on the level of noise.FIG. 23 shows a 5-link complex. The local behavior parameters are to befound at point t₀, (which, for simplicity, will be assumed to correspondto 0) given that sampled values of the signal are known for the next 10Nyquist rate sampling intervals. Oversampled data will be used only fromthe interval (-3,3) which is the union of the central intervals C₋₂,C₋₁, C₀, C₁, and C₂ of the five links Λ₋₂, Λ₋₁, Λ₀, Λ₁, and Λ₂. Outsidethe interval (-3,3) only the Nyquist rate samples of the signal areused. A quadratic expression, S, is constructed which will provide thedata needed to determine local behavior processing parameters. Consideran arbitrary link Λ_(i) from the complex. The following expressions areformed for each link Λ_(i).

1. The error of interpolation: the sum of the squares of the differencesbetween the interpolated values and the oversampled values of thesignal, using the interpolation function of Λ_(i) parametrized by theinternal values v_(j) ^(i) of the signal for the link Λ_(i) withparameters v_(j) ^(i) for all integers j between -k and k. ##EQU71##where [i-1,1) is closed on the left and open on the right, and [i,i+1]is closed on both the right and left.

Here the summation ##EQU72## refers to the sum of all values of term(Q^(i) (τ)-f(τ))² when τ takes as values the oversampling points fromthe interval C_(i). The v^(i) _(j) are variables which are used toformulate the constraint equation for the complex. Recall that Q^(i) ₋₁is the interpolation polynomial for the left half of the centralinterval, and Q^(i) ₁ (τ) for its right half.

2. Deviation of the internal values v^(i) _(j) from the sampling valuesf(j) at the Nyquist rate sampling points, both windowed by the windowW(j) corresponding to the interpolation function of the links. Thus, thefollowing expression is formed: ##EQU73## For the Lagrangian link##EQU74## Thus, S_(i) ² expresses the total energy of the differencebetween the "internal" values of the signal, and the sampled values.This sum controls the low frequency drift. Thus, minimizing the sum##EQU75## would produce a trade off between the accuracy of the match ofoversampled data, (and thus the highest end of the spectrum) and the lowfrequency drift. The possibility to find an arbitrarily close leastsquare fit at the expense of accumulating the low frequency drift isinsured by the Finite Local ε-base Theorem.

However, the central interval of a single link is not sufficient toinsure an accurate match of oversampled data in the presence of noisewith a lower degree of oversampling. This is the reason why a complex isused. The next condition insures that any two consecutive links arehighly consistent in terms of the internal values of the signal.Consider Λ_(i) and Λ_(i+1). The following sums are formed: ##EQU76##Where V(j)=(n!(n-1)!)/((n-j)!(n+j)! is the windowing function for L^(2k)(t) since the two consecutive links, each having an odd number ofpoints, intersect over an even support. Thus, S^(i) ₄ is an appropriatemeasurement of the deviation of the successive internal versions of thesignal in the two consecutive links, appropriately adjusted due to thefact that for each such value the windowing is different in the twolinks. The fact that appropriately weighted sum of squares of thesedifferences, i.e. ##EQU77## can be made small can be deduced fromconsiderations leading to the proof of the Finite Local ε-base Theorem.This condition controls the sizes of predictions when a complex is usedclose to the end of the data stream.

S^(i) ₃ is a measure of differences of the interpolations by the linkΛ_(i) and Λ_(i+1). A more realistic measure would be the integral overthe intersecting interval, but this one is more convenient andsufficiently accurate. This condition controls the low-frequency errorof the procedure.

Condition S^(i) ₄ is important because it ensures that at any centralpoint between the two consecutive Nyquist rate points, the differencesbetween the chromatic derivatives are small. In this way local the"spectral features" i.e. the values of chromatic derivatives become moreglobal, which ensures the consistencies of the "local spectra." Forsmall intervals there are many aliases having the same chromaticderivatives, but if these values are forced into consistency over longerperiods of time, this constrains what signals satisfy these conditionsto those signals whose upper parts of the Fourier transform have minimaldifferences. In this way over longer periods of time the spectra of thereplica of the signal and the signal will closely resemble each other.

In some applications requiring the highest accuracy in a particular partof the spectrum, condition S^(i) ₃ can be replaced by the more generalcondition of the form ##EQU78## where ##EQU79## is chosen so that it hasparticular frequency selection features. The time is measured"externally" as for the signal f(i).

In practice it is enough take operators involving chromatic derivativesup to the order 3-7 to determine with small error the form of the signalbetween any two consecutive sampling points.

The weighted sum ##EQU80## where i ranges over indices of all links ofthe complex is a consistency measurement. The orders of magnitude of theweights involved should reflect the error intrinsic to the correspondingquantity and the nature of approximation, but there is also a lot ofspace for adaptive or just tunable algorithms.

Having formed the above quadratic expression, the expression can beminimized in the usual manner. Thus, partial derivatives with respect tothe variables involved are taken and the corresponding system of linearequations is solved. This in fact produces a large system which shouldbe generated and solved by a symbolic manipulation program likeMathematica. Only the values of v_(j) for the focal point of the linkhave to be found as linear combinations of the Nyquist rate samples andoversampled values of the signal, and they can be the output of thecomplex. The coefficients of these linear combinations are then used inthe transversal filter structure as illustrated in FIG. 13.

The number of oversampled taps on the transversal filter for each v_(j)corresponds to the number of oversampling points in the union of thecentral regions (thick central part of the longer line denoting thesupport of each link) and the Nyquist rate taps correspond to the unionof all supports of the links.

Note that the above equations are solved off-line and any parameters(weights) are substituted to obtain expressions with constantcoefficients to be used in real time.

If the signal is noisy, one combine an integration procedure can befollowed by the above procedure with immediate differentiation of theapproximation function at the focal point, thus obtaining a highly noiserobust filter.

For some applications the output of the complex can be replaced by asequence of values of chromatic derivatives at the focal point.

The mode described above satisfies the property that for every point inthe support of each link there exists a sampled value of the signal, andis called the fully grounded mode. If the delay allowed is notsufficient that for every point v^(i) _(j) in the support of each linkΛ_(i) there corresponds a sampled value f(i+j), then the complexoperates in a predictive mode. The most complex case is if k points(assuming that each link has 2k+1 points in its support) of the lastlink do not have the corresponding sampled values, i.e. such values are"suspended" and related to the sampled values of the signal only throughthe constraints described above. This mode will be called the fullypredictive mode. Intermediate cases in which less than k points in thesupport of the link do not have the sampled values of the signalassociated with them is analogous to the case of a fully predictivecomplex, and are treated accordingly. An intermediate complex isillustrated in FIG. 24 and a fully predictive complex is illustrated inFIG. 25.

In the case of a fully predictive complex the second sum is of the form##EQU81## thus, the sum is truncated up to 0. To prevent instability ofthe corresponding system of linear equations obtained afterdifferentiation, the number of links must be increased, so that thereare several fully grounded links. An optimum must be found between theround-off error accumulation and numerical stability of the system ofequations. If necessary, other constraints can be added to limit theenergy in the suspended part of the last link: ##EQU82##

Finally, in the presence of a high degree of noise, integration,followed by the above, can be combined with differentiation of theoutput of the last link.

IX. Signal Processing Engine

FIG. 26 is a block diagram illustrating the top level components of anembodiment of a signal processing engine 200 according to a first aspectof the invention.

The signal processing engine 200 includes a data acquisition unit 310and a local signal behavior descriptor 410. The data acquisition unit310 accepts the input signal 210, sampling the input signal 210 at arate which is n times the band limit of the signal, where n is greaterthan 2, and outputs a digital representation 220 of the input signal 210to the local signal behavior descriptor 410. The local signal behaviordescriptor 410 calculates and outputs polynomial approximations 230.Depending on the particular application of the signal processing engine200, the output of the local signal behavior descriptor 410 may be inputto a processor which applies a second set of linear operators to extractinformation from the polynomial approximation 230 in order to provide aprocessed signal or to allow decisions to be made, e.g. pulse widthmodulator switching, or other applications.

The processing may include, for example, finding values of differentialoperators solving certain differential equations, or interpolation andextrapolation for the case of a fully predictive complex. Suchpredictions become increasingly inaccurate as the prediction point movesto the right; their main purpose is to define local signal behavior atthe end of the data stream, or extrapolations over a short period oftime. The length of this period depends on the allowed error. Theoperations applied are the standard polynomialinterpolation/extrapolation as provided by the output of the complex.

FIG. 27 is a block diagram illustrating the components of an embodimentof a data acquisition unit 310 of a signal processing engine accordingto a first aspect of the invention. FIG. 30 is a block diagramillustrating the components of a second embodiment of a data acquisitionunit 310' of a signal processing engine according to a first aspect ofthe invention. The block diagram in FIG. 27 represents a basic, oridealized, data acquisition unit 310 according to the invention. Theblock diagram in FIG. 30 represents the basic data acquisition unit 310augmented with certain additional features for responding to start-uptransients and other transients which may occur during operation of thesignal processing engine 200. The embodiment illustrated in FIG. 27 willbe described first, and then the embodiment illustrated in FIG. 30,illustrating primarily the differences between the two embodiments. Thecomponents in both FIGS. 27 and 30 which serve the same function in bothembodiments have the same reference numerals with the addition of a "'"("prime") to the corresponding reference numerals in FIG. 30.

Referring now to FIG. 27, within the data acquisition unit 310, theinput signal 210 is input to a non-inverting input 314 of a summingamplifier 316 (e.g., differential amplifier). The output of the summingamplifier 316 is coupled to the input of a programmable gain amplifier322, and the output of the programmable gain amplifier 322 is coupled tothe input of an analog to digital (A/D) converter 332.

The gain of the programmable gain amplifier 322 is controlled by a gaincontrol signal 243 provided by a control logic 350 module within thedata acquisition unit 310. The control logic 350 also provides avariable reference voltage signal 241 to the A/D converter 332. Thevariable reference voltage signal 241 is used to set the full scalerange of the A/D converter 332. Through a procedure (described below) bywhich the gain of the programmable gain amplifier 322 and the full scalerange of the A/D converter 332 are adjusted, the resolution of the A/Dconverter 332 is increased, i.e., the full scale of the A/D converterrange 332 is set up to measure small changes in the input signal 210.

The A/D converter 332 digital output 212 represents a high resolutionmeasurement of the difference between a predicted value of the inputsignal 210 and the actual value of the input signal 210. The A/Dconverter 332 digital output 212 is input to a scale adjust module 336.The scale adjust module 336 rescales the A/D converter 332 digitaloutput 212 back to the full scale of the input signal 210, whilepreserving the high resolution of the A/D Converter 332 measurement ofthe difference between the predicted value of the input signal 210 andthe actual value of the input signal 210. The scale adjust module 336receives a scale adjust control signal 244 from the control logic 350.The scale adjust control signal 244 sets up parameters in the scaleadjust module 336 corresponding to the gain of the programmable gainamplifier 322 and the full scale reference in the A/D converter 332.

The output of the scale adjust module output 336 is coupled to one inputof an adder 344 and also fed back to the control logic 350. The outputof the adder 344 is coupled to the input of a prediction filter 380(which may also be referred to as an extrapolation filter). Theprediction filter output 214 is coupled to the input of a digital toanalog (D/A) converter 338 and to a second input of the adder 344. Thus,the output of the adder 344 is the sum of the output of the predictionfilter 344, which is a predicted value of the input signal 210, and theoutput of the scale adjust module 336, which represents the differencebetween the predicted value of the input signal 210 and the actual valueof the input signal 210. Therefore, the output of the adder 344 is ahigh resolution measurement of the input signal 210. This highresolution measurement of the input signal 210, output from the adder344, is the digital representation 220 which is output from the dataacquisition unit 310 to the local signal behavior descriptor 410.

The output of the D/A converter 338 is coupled to an inverting input 318of the differential amplifier 316. Thus, the output of the differentialamplifier 316, and the signal which is passed through the programmablegain amplifier 322 to be sampled by the A/D converter 332, is thedifference between the current value of the input signal 210 and apredicted value 214, 215 of the input signal. This is described indetail below.

The local signal behavior descriptor 410 receives the digitalrepresentation 220 from the data acquisition unit 310, and calculatesthe outputs of linear operators applied to a polynomial approximation ofthe digital representation 220 of the sampled input signal 210. Thestructure and operation of the local signal behavior descriptor 410 isdescribed below.

FIG. 28 is a block diagram illustrating the components of an embodimentof a prediction filter 110 of a signal processing engine according to afirst aspect of the invention. The prediction filter is a transversalfilter corresponding to a fully predictive complex with coefficientsdetermined in the following manner:

The output of a fully predictive complex is of the form v⁰ _(-k), . . ., v⁰ ₀, . . . , v⁰ _(k), where the complex consists of links Λ_(-m), . .. , Λ₀, with the present time t₀ as the focal point of the link Λ₀.

Let P(t,v) (bold v denoting a vector quantity) be the approximationpolynomial corresponding to the link Λ₀. Then the predicted value b(t)is found as ##EQU83## Here, t₁ is the first oversampling point followingt₀. Since all of v⁰ _(-k), . . . , v⁰ ₀, . . . , v⁰ _(k) are linearcombinations of sampled values of the signal as provided by the complex,##EQU84##

The coefficients c_(m) for Nyquist rate samples f(m), and d.sub.τ forthe oversampling rate values of f, i.e. f(τ), are obtained from theequation for b(t) above by substituting for v⁰ _(j) with thecorresponding linear combinations for v_(i) as given by the equationsresulting from the complex. Since there are no Nyquist rate points andno oversampling rate points to the right of t₀, the resulting linearcombination ##EQU85## is realized by a transversal filter of the formillustrated in FIG. 36, if the complex used is of the form illustratedin FIG. 25. The transversal filter includes 7 Nyquist rate taps and11×6=66 taps on the oversampled scale, corresponding to the sampledvalues in the union of the central regions of all links.

FIGS. 29A and 29B are flow charts which illustrate aspects of theoperation of an embodiment of a control logic module of a signalprocessing engine according to a first aspect of the invention. The flowchart of FIG. 29A illustrates an initialization process 352 for the dataacquisition unit 210. The flow chart of FIG. 29B illustrates a processfor increasing the resolution of the A/D converter 332 during normaloperation of the signal processing engine 200.

Referring to FIGS. 27, 28 and 29A, the operation of the data acquisitionunit 310 proceeds as follows. At start-up of the signal processingengine 200, the control logic 350 initializes the data acquisition unit310, setting all sample values to zero. The predicted value 214 outputfrom the prediction filter 380 is set to zero, the programmable gainamplifier 322 is set to unit gain, and the reference voltage for the A/Dconverter 332 is set to its highest level. This level is chosen so thatthe full scale of the D/A converter 338 corresponds to the full scale ofthe A/D converter 332, i.e. the maximum amplitude of the input signal210.

The A/D converter 332 begins sampling the input signal 210. After theA/D converter 332 has sampled n samples, at sampling points t=a₁, a₂, .. . a_(n), and the corresponding sampled values of the input signalf(a₁), . . . f(a_(n)) are obtained, the prediction filter 380 calculatesa predicted value P(a_(n+1)) for the next sampling point t=a_(n+1). Thisvalue is sent from the prediction filter 380 to the D/A converter 338.The D/A converter 338 converts the predicted value for the next samplingpoint to analog, and the converted predicted value for the next samplingpoint is input to the differential amplifier 316 where it is subtractedfrom the current value of the input signal 210.

The total error of the prediction filter 380 is the sum of the round offerror which comes from the quantization error of the A/D converter 332and the error of the polynomial approximation of the prediction filter380. The resolution of the A/D converter 332 and the degree of thepolynomial approximation of the prediction filter 380 are chosen so thatthe total sum of these two errors is smaller than a fraction 1/2^(s)(for some s≧1) of the full scale of the input. Thus, the differencevoltage at the output of the differential amplifier 316 will be smallerthan 1/2^(s) of the full scale of the input. FIG. 30 illustrates whypolynomial approximations reduce the size of a signal which is to besampled, i.e., instead of the full signal, the output of thedifferential amplifier 316 represents the approximation error from theprediction filter 380 combined with the quantization error of the A/Dconverter 332.

Referring again to FIGS. 27, 28 and 29A, for the next sample, i.e., att=a_(n+2), the gain of the programmable gain amplifier 316 and the fullscale reference of the A/D converter 332 are adjusted to set the fullscale of the A/D converter 332 to 1/2^(s) of the full scale of the inputsignal 210. This increases the resolution of the A/D converter 332, andhence the digital representation 220, by s bits. After n more samples,the resolution adjustment procedure is repeated and the resolution isagain increased by s bits. Thus, each time the resolution is improved,the size of the quantization error is reduced, thereby allowing a higherresolution.

After a number of cycles, a "baseline" resolution, is achieved. The dataacquisition unit 310 finishes the initialization procedure, and thesignal processing engine 200 begins normal operation. The number ofcycles the initialization procedure takes to establish the baselineresolution may be a predetermined number of cycles, or the procedure mayrun until a desired baseline resolution is achieved.

The resolution achieved during the initialization procedure is referredto as a baseline resolution because the resolution may also be increasedduring normal operation of the engine. Referring to FIG. 29B, theresolution may be increased in the following manner. The control logic350 continually monitors the output of the scale adjust module 336. Ifthe output of the scale adjust module 336 is less than a predeterminedvalue for a specified number of consecutive samples, then the controllogic 350 causes the gain of the programmable gain amplifier 316 and/orthe full scale reference of the A/D converter 332 to be adjusted toagain set the full scale of the A/D converter 332 to 1/2^(s) of the fullscale of the input signal 210. The scale adjust module 336 is alsoupdated to match any changes made to the gain of the programmable gainamplifier 316 and/or the full scale reference of the A/D converter 332.Although not illustrated in the FIGs., the control logic 350 may also beconfigured to reduce the resolution of the A/D converter 332 byadjusting the gain of the programmable gain amplifier 316 and/or thefull scale reference of the A/D converter 332, if the output of thescale adjust module 336 exceeds specified criteria.

The topology of the data acquisition unit 310 differs from the usual"pipeline" architecture of A/D converters in two fundamental ways.First, the predicted value of the input signal is known well in advancesince the predicted value is produced by simple computations from thevalues of the signal at previous sampling points. Thus, the D/Aconverter 338 has an ample amount of time to settle, and higherprecision can be achieved much easier. Second, the procedure isdependent on the spectral content of the signal which makes samplingadaptive, and maintains a high resolution only when justified by theparticular application, and is controlled entirely by parametersobtained internally with relatively simple computations.

During normal operation, the signal processing engine 200 may continueto increase its resolution whenever the approximation error is low.However, increased sampling resolution may not be immediately reflectedin the number of digits used to code the sample values in the output ofthe signal processing engine 200. Instead, the increased resolutionsharply reduces the quantization (round-off) error, making the intrinsicerror of the polynomial approximation the larger contributor to thetotal error. The prediction error is averaged over a reasonably longperiod of time. Generally, the total error E(t) of a polynomialextrapolation is the sum of the error of the polynomial approximationitself and the cumulative effect of the round-off error. As theround-off error, caused by quantization, decreases sharply withincreased resolution, the total error becomes approximately equal to theerror of the approximation E_(app) (t)=|f(t)-Y(t)|. However, in general,the error of polynomial approximations can be bounded by a bound whichdepends on the value of a certain derivative of the signal, the order ofthe derivative depending on the number of points in the approximation.This bound is expressible in terms of the energy of the signal and ishighly frequency-dependent. This is due to the filtering characteristicsof derivation. The average value of error reflects the energy oftransients. Thus, a suitable look-up table can be used to calculate thetiming for increasing resolution according to the effect of the maskingphenomena.

The maximal possible increase in resolution is eventually limited by thelimitations imposed by the noise of the hardware.

As indicated above, the embodiment of a data acquisition unit 310illustrated in FIG. 27, is a basic, or idealized, data acquisition unit.The block diagram in FIG. 30 represents a data acquisition unit 310'augmented with certain additional features for responding to start-uptransients and other transients which may occur during operation of thesignal processing engine.

Referring now to FIG. 30, within the data acquisition unit 310', theinput signal 210 is input to a soft-start window 312. The purpose of thesoft-start window 312 is to dampen initialization transients uponstart-up of the signal processing engine 200, and particularly in orderto facilitate the initialization of the prediction filter 380'. FIG. 31is a graph which illustrates the transfer function of the soft-startwindow 312 as a function of time, and shows that the soft-start window312 simply acts as a variable attenuator. Upon start-up, the output ofthe soft-start window 312 is initially forced to zero, i.e. 100 percentattenuation. After a brief delay, the attenuation of the soft-startwindow 312 is reduced, thereby permitting the output range to increase.When the attenuation reaches zero, the soft-start window 312 appears asa switch in the closed position. The length of the delay prior toreducing the attenuation, and the rate at which the attenuation isreduced will depend on the particular application of the signalprocessing engine 200.

The soft-start output 211 is coupled to a non-inverting input 314' ofthe differential amplifier 316'. The output of the differentialamplifier 316' is coupled to the input of the programmable gainamplifier 322', and the output of the programmable gain amplifier 322'is coupled to a normally closed (N.C.) position 324 of an A/D inputselection switch 328. The soft-start output 211 is also directly coupledto a normally open (N.O.) position 326 of the A/D input selection switch328. The A/D input selection switch 328 is coupled to the input of theA/D converter 332'.

The gain of the programmable gain amplifier 322' is controlled by thegain control signal 243' provided by the control logic 350' module. Thecontrol logic 350' also provides the variable reference voltage signal241' to the A/D converter 332'. The variable reference voltage signal241' is used to set the full scale input range of the A/D converter332'.

The A/D converter 332' digital output 212' is input to the scale adjustmodule 336'. The scale adjust module 336' receives a scale adjustcontrol signal 244' from the control logic 350'. The scale adjustcontrol signal 244' sets up parameters in the scale adjust module 336'corresponding to the gain of the programmable gain amplifier 322' andthe full scale reference in the A/D converter 332'.

The output of the scale adjust module 336' is coupled to a first inputof the adder 344'. The output of the adder 344' is coupled to theprediction filter 380', and the output of the prediction filter 380' iscoupled to the input of the D/A converter 338' and to a D/A calibrationtable 342. The output of the D/A calibration table 242 is coupled to asecond input of the adder 344'. The output of the D/A converter 338' iscoupled to an inverting input 318' of the differential amplifier 316'.

The differences between the basic data acquisition unit 310 illustratedin FIG. 27, and the augmented data acquisition unit 310' illustrated inFIG. 31, include the soft-start window 312, the A/D input selectionswitch 328 with the direct coupling from the output of the soft-startwindow 312, the D/A calibration table 342, and certain transientrecovery functions performed by the control logic 350'.

The function of the soft-start window 312 is described above. The D/Acalibration table 342 is a table which calibrates the output of the D/Aconverter 338' against an ideal D/A converter. It is important that thevalue which is input to the adder 344' accurately represent the valuewhich is actually output by the D/A converter 338', i.e., including anyerror inherent in the D/A converter 338', and input to the differentialamplifier 316'. Therefore, the D/A calibration table 342 inputs thepredicted value 214' from the prediction filter 380', and feeds back tothe adder 344' a digital value which represents the analog value outputfrom the D/A converter 338'.

The flow chart in FIG. 32 illustrates the transient recovery operationof the control logic 350'. Referring briefly to FIG. 30, the controllogic 350' monitors an A/D in range signal 213 from the A/D converter332'. Under normal operation, the A/D in range signal 213 shouldindicate that the signal input to the A/D converter 332' from theprogrammable gain amplifier 322' is within the range of the A/Dconverter 332'. However, if, for example, due to a transient in theinput signal 210', the signal input to the A/D converter 332' saturatesthe A/D converter 332' and exceeds the full scale range of the A/Dconverter 332', the control logic 350' sends a scale adjust controlsignal 244' to the scale adjust module 336' to ignore the current samplefrom the A/D converter 332', and to use the current predicted value asthe actual value for the current sample. The control logic 350' alsosends an A/D input switch signal 242 to the A/D input selection switch328 causing the A/D input selection switch 328 to switch to the N.O.position 326, thereby coupling the output of the soft-start window 312directly to the input of the A/D converter 332'. The control logic 350'also sends a variable reference voltage signal 241' to reset the fullscale range of the A/D converter 332', a gain control signal 243' toreset the gain of the programmable gain amplifier 322' to unity, and ascale adjust control signal 244' to reset the scale adjust module 336.

On a subsequent sample, the A/D in range signal 213 will indicate thatthe signal input to the A/D converter 332' is within the range of theA/D converter 332'. The control logic 350' then tests to determine theposition of the A/D input selection switch 328. The first N timesthrough this path, N representing a predetermined period to allowsettling of a transient on the input, the A/D input selection switch 328will be in the N.O. position 326 having been set when the out of rangecondition was first detected. After the signal input to the A/Dconverter 332' is within the full scale range of the A/D converter 332'for N consecutive samples, the control logic 350' sends an A/D inputswitch signal 242 to the A/D input selection switch 328 causing the A/Dinput selection switch 328 to switch to the N.C. position 324, therebycoupling the output of the soft-start window 312 to the input of the A/Dconverter 332' through the differential amplifier 316' and programmablegain amplifier 322'. The control logic 350' then starts theinitialization process illustrated by the flow chart of FIG. 29A.

The output of the polynomial approximator in the data acquisition unitis a (2k+1) tuple of values of the complex implemented in the dataacquisition unit. Thus, it might seem that the oversampled data was notcompressed since each Nyquist rate sampling point produced (2k+1) manyvalues which can be substantially more than the degree of oversampling.However, this is not the case, since in processing data at each samplingpoint, only these 2k+1 points are used, due to the fact that alloperators are defined from locally supported operators (recursioninvolving the previous value of computation can be used, but not any newsampling points.)

Thus, any filtering computes the value from the previous value of thefiltered signal using only 2k+1 (e.g. 17 in this embodiment) new localbehavior parameters associated with the new point.

The derivation of local behavior parameters can be achieved using a"smart" LSB based A/D converter using, for example, a CCD implementedtransversal filter.

Once the configuration of a transversal filter for an application hasbeen determined, including number of Nyquist rate taps, number ofoversampled rate taps, coefficients represented by the amplifiers, etc.,the implementation of the filter may be "hardwired," i.e. it need not beprogrammable because the configuration and coefficients will not change.Also, a processor implementing the data acquisition unit can useoversampled data not only to obtain the local behavior parameters, butalso to predict the new value of the signal and thus significantlyreduce the dynamic range of the A/D converter. This prediction isobtained by evaluating the approximation function of the last link atthe point in the future corresponding to the next sampling point.

Within the processor the basic calculations (differentiation,integration, solution of differential equations etc) are performed usingan interpolation with 2k+1 points, or just 4-7 values of the chromaticderivatives.

Implementations of the LSB method and engine of the invention can belayered with significantly lower Nyquist rates, correcting the lowfrequency error of the primary stage. Also, for example, filtering canbe implemented in a mix of LSB plus harmonic technologies. For example,FIG. 34 is a diagram which illustrates an arrangement 140 which includesa combination of an LSB Engine 142 and a harmonic low pass filter 144.In the arrangement 140 illustrated in FIG. 34, a signal is input to boththe LSB Engine 142 and to a non-inverting input of a unit gaindifferential amplifier 143. The output of the LSB Engine 142 is coupledto an inverting input of the unit gain differential amplifier 143 and toa first input of an adder 145. The output of the unit gain differentialamplifier 143 is coupled to the input of the harmonic low pass filter144, and the output of the harmonic low pass filter 144 is coupled to asecond input of the adder 145.

The cut off frequency of the harmonic low pass filter 144 is chosen tobe sufficiently within the pass band of the LSB Engine 142. The harmoniclow pass filter 144 does not "see" any frequencies near its cut-offfrequency, and therefore it introduces no significant phase shift. Therole of the harmonic low pass filter 144 is confined to isolating thelow frequency error of the LSB Engine 142. This arrangement will beillustrated in the following description of a switch mode amplifierwhich employs the LSB signal processing method and engine of theinvention.

X. Switch Mode Amplifier

FIG. 35 is a diagram of an embodiment of a switch mode power amplifier500 employing the signal processing method and engine of the invention.The application of the signal processing method and engine of theinvention to the design of a switch mode power amplifier overcomesshortcomings of existing switching amplifiers, e.g. class "D"amplifiers. These shortcomings include: poor handling of highly reactivecomplex loads (e.g., speakers), usually requiring a duty cycle orfeed-back adjustment with the change of the load; poor performance inthe upper part of the bandwidth, including numerous switching artifacts;and high distortion, especially in the upper part of the spectrum. Theseshortcomings are all overcome using the local signal behavior signalprocessing method and engine of the invention.

Referring first to FIG. 36, illustrated is a circuit diagram of certaincomponents of the switch mode amplifier 500 illustrated in FIG. 35. Thearrangement includes a pulse width modulator (PWM) 502 which is coupledto a switching regulator 504. The switching regulator 504 may be apush-pull or other switching regulator known to those skilled in theart. The output of the switching regulator 504 is coupled to a 10 kHzfilter 511 which includes a series inductor 510 and a parallel capacitorC1 512. The output of the 10 kHz filter 511 is coupled to a reactiveload 515. It can be shown that, in the circuit of FIG. 36,

    V.sub.1 (t)=L.sub.1 C.sub.1 V.sub.in.sup.(2) (t)+L.sub.1 i.sub.o '(t)+V.sub.in (t).

Using elementary relationships between the voltages and currents, andthe derivatives of the voltages and currents in the circuit of FIG. 36,it can be established that, in order that at any time t the voltageV_(o) (t) on the output of the circuit is equal to a given voltageV_(in) (where V_(in) (t) is the input voltage multiplied by the voltagegain), the voltage V₁ (t) must be equal to the value of the followingdifferential operator:

    D(V.sub.in, i.sub.o)=L.sub.1 C.sub.1 V.sub.in.sup.(2) (t)+L.sub.1 i.sub.o '(t)+V.sub.in (t)

Thus, the voltage V₁ (t)=D(V_(in), i_(o))(t) depends on the inputvoltage V_(in) (t) and the output current i_(o) (t). Here, i_(o) (t) istreated as a fully independent parameter and this allows the amplifierto drive any reactive load, since no assumption is being made about therelationships between the output voltage V_(o) (t) and the outputcurrent i_(o) (t). In this manner, the output voltage of the circuit isalways in perfect phase with V_(in) (t). Thus, the differential operatorD(V_(in), i_(o)) essentially performs reverse filtering. This allows asmall linear power amplifier to correct any error made by the switchingcircuit, by simple comparison of the voltages V_(o) (t) and V_(in) (t).

The L₁ C₁ network 510, 512 is chosen so that the cut-off frequency ofthe filter 511 is well within the audible spectrum (e.g., 10 kHz). Thisstrongly suppresses the switching noise and the high frequency artifactsof switching, leaving for the linear amplifier an easy task ofcorrecting the remaining low level error in the presence of a very lowswitching noise.

Referring again to FIG. 35, the details of operation of the amplifiercircuit are described below. The control unit of the amplifierexemplifies applications of the signal processing method and engine ofthe invention in control systems in general, and also how harmonic meanscan be used in conjunction with the LSB engine. Also, despite theobvious fact that the design can be realized purely in digitaltechnology, those skilled in the art will recognize that the embodimentillustrated can also be conveniently realized in CCD technology.

Since the value D(t)=L₁ C₁ V_(in).sup.(2) (t)+L₁ i_(o) '(t)+V_(in) (t)involves values of inductors and capacitors which may be known withlimited accuracy and which are subject to change over time due, forexample, to aging, , and due to the intrinsic error of the signalprocessing used to determine V_(in).sup.(2) (t) and i_(o) '(t), theamplifier must rely on the feed back to ensure the accuracy of V_(o).Since the switching frequency is only 5 times the Nyquist rate, thenumber of recursive iterations in the feedback loop due to the digitaldelay (action can be taken only once per switching cycle) is very low.

The amplifier circuit includes the switching regulator 504, a 10 kHz LCfilter 511, a low power linear amplifier 524 and two feedback loops. Aninner feedback loop 546 includes an LSB Engine 1 532 with weights"tuned" maximally toward the upper part of the spectrum, allowingsubstantial error (drift) in the low end of the spectrum. This meansthat S² _(i) has low weight allowing that parameters V^(i) _(j), departsignificantly from the values f(i+j), while differential operators arekept tight from one link to another link. In this way, the output of LSBEngine 1 532 is of the form S_(A) +S_(L), where S_(A) is the signal (theaudible part of the spectrum of the voltage V₁) and S_(L) is the lowfrequency error signal produced by LSB Engine 1 532.

The voltage V₁ contains also the high frequency noise S_(N). Bysubtracting the output of LSB Engine 1 532 from V₁ =S_(A) +S_(N), thesignal S_(N) -S_(L) is obtained, which is then filtered by a harmonicfilter F1 528. However, the input to harmonic filter F1 528 does nothave any significant component around its cutoff frequency of about12-18 kHz. Thus, no significant delay or phase shift is produced, andthe output is S_(L), with negligible delay and phase shift.

An adder 530 adds the output of the harmonic filter F1 528 to the outputof LSB Engine 1 532, and provides S_(A) with no phase shift or delay.This provides the feedback voltage for the pulse width modulator 502.

The noise on the input of LSB Engine 1 532 is very high. Thus, LSBEngine 1 532 comprises a Σ-Δ filter as illustrated in FIG. 37,preferably with an analog leaky integrator as illustrated in FIG. 38.Integration is done on an "infinitely fine" scale by analog integration,and differentiation is performed by LSB Engine 1 532 to which the analogleaky integrator sends its output. In FIG. 38, the analog leakyintegrator 150 is shown coupled directly to the input of the transversalfilter 120 of FIG. 37. Although not illustrated, a data acquisition unit310, as illustrated in FIGS. 26, 27 and 30 would be coupled between theleaky integrator 150 and the transversal filter 120. The low end of thespectrum of the error is corrected by the harmonic filter F1 528.

The driving signal for the amplifier 500 is provided by an LSB Engine 4522, producing on its output V_(in) and L₁ C₁ V_(in).sup.(2) +V_(in).V_(in) is only delayed by LSB Engine 4 522. This filter has the easiesttask since it provides V_(in).sup.(2) in conditions of high oversampling(the delay chain can be made with, for example, 4 μsec delay unitscorresponding to about 5× oversampling). The input signal is essentiallynoise free and small delay <1 msec introduced by this filter isinconsequential. Thus, LSB Engine 4 522 provides a fully groundedcomplex. LSB Engine 4 522 comprises a transversal filter 130 asillustrated in FIG. 38.

The input for the PWM 502 is provided by a differential operator 520

    D=L.sub.1 C.sub.1 V.sub.in.sup.(2) +L.sub.1 i.sub.o '+V.sub.in

with i_(o) ' being altered by the outer feedback loop 544.

The operation of the outer feedback loop 544 is as follows. The linearamplifier 524 maintains the output voltage to the exact level V_(in),and the correction current, i_(c), supplied by the linear amplifier 524is sensed and processed by LSB Engine 2 540 which provides i_(c) andi_(c) '. The current i_(c) is maintained to a small fraction, typicallyone percent, of the output current i_(o).

Any deviation from this nominal value is compensated for by the outerfeedback loop 544 operating as follows. The value i_(o) ' is altered intwo ways. First, to insure tracking and the prevent the error build up,i_(o) ' is corrected for the value of the derivative of the correctioncurrent i_(c) '. This prevents the error accumulation. The reduction byfurther error correction for the factor ##EQU86## where ∥i_(o) ∥=|i_(o)| for |i_(o) |>c

=c for |i_(o) |≦c

i.e., ∥i_(o) ∥ is equal to the absolute value of the current i_(o), andis set to a small constant, c, if the absolute value of i_(o) dropsbelow c.

The coefficient α is chosen so that α is the quotient determining thenominal value of i_(c) compared with the value of the output current.The coefficient β corresponds to the feedback slope since it determinesthe rate of convergence of the correction. Altering i_(o) ' in such away to obtain i_(cor) ' essentially factors in the compensation for theerror with a stable convergence rate and prevents the further buildup ofthe current error.

LSB Engine 2 540 comprises a filter 120 as illustrated in FIG. 37. LSBEngine 3 538 has an input with much smaller ripple that does LSB Engine1 532. No Σ-Δ filtering is needed for LSB Engine 3 538. Thus, LSB Engine3 538 is a differentiator operating on the end of the data stream, andis a fully predictive complex as illustrated in the graph of FIG. 25.LSB Engine 3 538 is also implemented as a transversal filter 120illustrated in FIG. 37.

In reference to the description of the switch mode amplifier 500illustrated in FIG. 35, the descriptions of LSB Engine 1 532, LSB Engine2 540, LSB Engine 3 538 and LSB Engine 4 522 indicate the configurationsof transversal filters which may be used in an implementation of eachrespective LSB engine 532, 540, 538, 522. The transversal filter in eachrespective LSB engine 532, 540, 538, 522 corresponds to a polynomialapproximator within the local signal behavior descriptor 410 illustratedin FIG. 33. In particular implementations of the switch mode amplifier500, each of the LSB engines 532, 540, 538, 522 may have its ownrespective data acquisition unit 310 (illustrated in FIGS. 26, 27 and30), or the four LSB engines 532, 540, 538, 522 may share a singlemultiplexed data acquisition unit 310.

XI. Summary: Harmonic Analysis and Local Signal Behavior Processing

FIGS. 40, 41 and 42 are diagrams which illustrate differences betweenthe approach used by harmonic analysis and the approach used by thelocal signal behavior processing method and engine of the invention.

As illustrated in FIG. 40, in order to find the value of the output of aprocedure at a point t₀, harmonic analysis uses a large number ofNyquist rate data points (specific example shown uses about 100), anduses the Nyquist rate sampled values f(i) directly in the computations.

As illustrated in FIG. 41, the local signal behavior processing methodand engine of the invention, instead, use a much smaller number ofNyquist rate data points, e.g. 12-24 (specific example shown uses16-20), and use oversampled data points from a short interval, e.g. a1-5 Nyquist rate interval (specific example shown uses 2-3).

As illustrated in FIG. 42, the local signal behavior processing methodand engine of the invention do not output the sampled values directly,but instead transform the sampled data points into local signaldescription parameters. Each point is assigned its respective localsignal description parameters, Processing at each point t₀ uses only thelocal signal description parameters assigned to to and previous resultsof the procedure, but does not use the values of other points or localsignal description parameters assigned to other points.

All operators of the signal processing method and engine of theinvention act locally, or are defined by recursion from operators whichact locally.

Therefore, in accordance with the invention, extraction of local signalbehavior description parameters is uniform, and once configured, needsno program changes. Implementations of the signal processing engine ofthe invention can be "hard wired."

It will be readily apparent to a person skilled in the art that numerousmodifications and variations of the present invention are possible inlight of the above teachings. It is therefore to be understood that,within the scope of the appended claims, the invention may be practicedother than as specifically describe.

I claim:
 1. A signal processor comprising data description means forcharacterizing a local behavior of a band limited signal, the datadescription means comprisingdata acquisition means for sampling thesignal at a rate which is n times the band limit of the signal, where nis greater than 2, and local signal behavior descriptor means forcalculating an output of a linear operator applied to a polynomialapproximation of the sampled signal.
 2. A signal processor according toclaim 1, wherein the linear operator comprises a locally supportedoperator.
 3. A signal processor according to claim 2, wherein thelocally supported operator comprisesa differential operator, an integraloperator, an interpolation operator, or an extrapolation operator.
 4. Asignal processor according to claim 3, wherein the locally supportedoperator comprises a differential operator, and the differentialoperator comprisesa first derivative, a second derivative, a thirdderivative, a fourth derivative, a fifth derivative, or a sixthderivative.
 5. A signal processor according to claim 4, wherein thedifferential operator comprises a chromatic derivative.
 6. A signalprocessor according to claim 1, wherein the linear operator comprises anoperator which is recursively defined from a locally supported operator.7. A signal processor according to claim 1, wherein the polynomialapproximation comprises a piecewise polynomial approximation.
 8. Asignal processor according to claim 1, wherein the polynomialapproximation comprises a Lagrange polynomial approximation.
 9. A signalprocessor according to claim 1, wherein the local behavior descriptormeans comprises a transversal filter.
 10. A signal processor accordingto claim 1, wherein the data acquisition means comprises analog todigital conversion means for sampling a signal and outputting a digitalrepresentation of the sampled signal.
 11. A signal processor accordingto claim 1, wherein the data acquisition means comprises soft-startmeans for dampening initialization transients upon start-up of thesignal processor.
 12. A signal processor according to claim 1, whereinthe data acquisition means has a full scale and a resolution, andcomprisesprediction means for providing a predicted value of the sampledsignal, differencing means for determining the difference between thesampled signal and the predicted value of the sampled signal, andadjustment means for adjusting the full scale and the resolution inresponse to the difference between the sampled signal and the predictedvalue of the sampled signal.
 13. A switch mode amplifier comprisinginputmeans for receiving an input voltage, a low pass filter for providing anoutput voltage and output current to a load a switching regulator forregulating a voltage input to the low pass filter, pulse widthmodulation control means for controlling the switching regulator,correction means for comparing the output voltage with the input voltageand supplying a correction current to the load based on the results ofthe comparison, an inner feedback loop for sensing a switching regulatoroutput voltage at an output of the switching regulator and providing aninner feedback loop input to the pulse width modulation control means,the inner feedback loop input responsive to the switching regulatoroutput voltage,the inner feedback loop comprising a first signalprocessor according to claim 1, the first signal processor adapted toinput the switching regulator output voltage and providing an audiblecomponent of the switching regulator output voltage, an outer feedbackloop for sensing the output current and the correction current andproviding an outer feedback loop input to the pulse width modulationcontrol means, the outer feedback loop input responsive to the outputcurrent and the correction current, the outer feedback loop comprisingasecond signal processor according to claim 1, the second signalprocessor adapted to input the sensed value of the correction currentand to output a first derivative of the correction current, and a thirdsignal processor according to claim 1, the third signal processoradapted to input the sensed value of the output current, and to output afirst derivative of the output current, and a fourth signal processoraccording to claim 1, the fourth signal processor adapted to receive theinput voltage and to output a second derivative of the input voltage tothe pulse width modulation control means.
 14. A signal processoraccording to claim 1, the signal processor comprising means for creatingone or more monads, each monad having its own interpolation polynomialand joining means for joining adjacent monads into a simplex or acomplex.
 15. A signal processor according to claim 14, wherein the meansfor linking adjacent monads comprises using least square fit proceduresto match polynomial approximations of adjacent monads.
 16. A signalprocessor according to claim 14 wherein the means for linking adjacentmonads comprises using least square fit procedures to match chromaticderivatives of adjacent monads.
 17. A signal processing method forcharacterizing a local behavior of a band limited signal, the signalprocessing method comprising the steps ofa. sampling the signal at arate which is n times the band limit of the signal, where n is greaterthan 2, b. calculating an output of a linear operator applied to apolynomial approximation of the sampled signal.
 18. A signal processingmethod according to claim 17, wherein the linear operator comprises alocally supported operator.
 19. A signal processing method according toclaim 18, wherein the locally supported operator comprisesa differentialoperator, an integral operator, an interpolation operator, or anextrapolation operator.
 20. A signal processing method according toclaim 19, wherein the locally supported operator comprises adifferential operator, and the differential operator comprisesa firstderivative, a second derivative, a third derivative, a fourthderivative, a fifth derivative, or a sixth derivative.
 21. A signalprocessing method according to claim 20, wherein the differentialoperator comprises a chromatic derivative.
 22. A signal processingmethod according to claim 17, wherein the linear operator comprises anoperator which is recursively defined from a locally supported operator.23. A signal processing method according to claim 17, wherein thepolynomial approximation comprises a piecewise polynomial approximation.24. A signal processing method according to claim 17, wherein thepolynomial approximation comprises a Lagrange polynomial approximation.25. A signal processing method according to claim 17, wherein the stepof sampling the signal comprises converting sampling an analog signaland converting the analog samples to a digital representation of thesampled signal.
 26. A signal processing method according to claim 17,wherein the step of sampling the signal comprises a soft-start step fordampening initialization transients.
 27. A signal processing methodaccording to claim 17, wherein the step of sampling the signalcomprisesestablishing a full scale and a resolution for sampling thesignal, calculating a predicted value of the sampled signal, determiningthe difference between the value of the sampled signal and the predictedvalue of the sampled signal, and adjusting the full scale and theresolution in response to the difference between the value of thesampled signal and the predicted value of the sampled signal.
 28. Asignal processing method according to claim 17, the method comprisingthe steps ofc. creating one or more monads each having it's owninterpolation polynomial; and d. joining adjacent monads together into asimplex or complex by using least square fit procedures to match thepolynomial approximations and chromatic derivatives of adjacent monads.29. A signal processing method for characterizing a local behavior of aband limited signal, the signal processing method comprising the stepsofa. sampling the signal at a rate which is m times the Nyquist rate forthe band limit of the signal, where m is greater than 1, every mthsample being a Nyquist rate sample, the time between two successiveNyquist rate samples being a Nyquist rate interval, and b. calculating alocal signal description parameter for a sampled value of the signal ata time t₀, the local signal description parameter comprising an outputof a linear operator applied to a polynomial approximation of the signalat the time t₀, the polynomial approximation comprising not more than 24Nyquist rate samples of the signal and all of the samples of the signalfrom not more than about 5 Nyquist rate intervals.
 30. A signalprocessing method according to claim 29, wherein the linear operatorcomprises a locally supported operator.
 31. A signal processing methodaccording to claim 30, wherein the locally supported operator comprisesadifferential operator, an integral operator, an interpolation operator,or an extrapolation operator.
 32. A signal processing method accordingto claim 29, wherein the linear operator comprises an operator which isrecursively defined from a locally supported operator.
 33. A signalprocessing method according to claim 29, wherein the polynomialapproximation comprises not more than 12 Nyquist rate samples of thesignal.
 34. A signal processing method according to claim 29, whereinthe polynomial approximation comprises not more than 1 Nyquist rateinterval.
 35. A signal processing method according claim 29, the methodcomprising the steps ofc. creating a monad at each Nyquist rate samplingpoint, each monad having its own interpolation polynomial, and d.joining adjacent monads together into a simplex or complex by usingleast square fit procedures to match the polynomial approximations andchromatic derivatives of adjacent monads.